Unit Conversions: The good, the bad and the ugly

Crutches

Like most chemistry teachers, one of the first things I go over in the beginning of the year is unit conversions. Students come into my class with all sorts of prior knowledge concerning unit conversions; some good, some bad and some downright ugly.

Every year I teach unit conversions I make some changes and I see progress. This year has been the most successful year yet with this topic. I write this post not as someone who has it all figured out but as someone who has seen the fruits of taking chances and making changes with my pedagogy. This post is a compilation of my observations of the prior knowledge students come in with and how I have dealt with them.

First, the good. I use Modeling Instruction pedagogy in my classroom so we do a lot of graphing and discussion of relationships between variables from day one. The good thing is that students come in knowing is how to deal with relationships. A student may give you a blank stare when asked to convert 6.00 meters to centimeters but if you tell that student that for every 1 meter, there are 100 centimeters, chances are that student will be able to answer the original question. Their brains are already wired to think proportionally.

In the beginning of the year, I give each group in my class a piece of cardstock that has been divided into 10 equal sections. Each group has a different length piece of cardstock. Students must measure the length, width and height of their desks with their new “ruler.” I even let them, make up the name for the units on their ruler. That gives us an opportunity to talk about precision and significant figures. Later, I give students the measurements of the length, width and height of their desks in my unit, the “stewart.” I also give students the width of my demo desk in stewarts. It is each group’s task to figure out the width of my demo desk in their units. When I went around to talk to each group this year, many groups explained that they had found the relationship between stewarts and their unit to solve the problem. Groups either set up a proportion between the units or used to the relationship they calculated to scale the measurements they already had. Either way, students were automatically thinking proportionally.

Now the bad. When I say bad, I mean something that is not inherently a bad practice but is bad when taught algorithmically. In this case, I am talking about dimensional analysis. In the beginning of my teaching career, I let students choose whether they wanted to use dimensional analysis or proportional reasoning. I knew student choice was valuable and I wanted students to be able to do what made sense to them. Good idea, not so great in practice. This ended up in a lot of what I call math monsters. A math monster occurs when a student tries to combine proportional reasoning AND dimensional analysis into one equation that makes no sense because there is either no variable to solve for or the other side of the equation is missing so they cannot solve for the variable. I realized that students were creating these math monsters because they were trying to repeat a set of steps they did not understand. Instead of providing students with multiple avenues to solve a problem, I had provided them with multiple algorithms to memorize and confuse.

Finally, the ugly. The ugly includes crutches that I have increasingly seen students use that have hindered their abilities to actually understand unit conversions. I will be the first to admit that I have been both a victim and perpetuator of some of these crutches.

The first "ugly" thing I have seen students do is decimal hopping to convert between metric units. If you don’t know what I’m talking about, it looks like this:

You and I understand that for every 1 meter, there are 100 centimeters, so moving the decimal over 2 spaces is essentially multiplying by 100 cm/m. Students do not see that. They see a shortcut that gets them to right answer without understanding the process.

The other ugly thing I have seen is magic triangles, specifically in regards to density (though they can be used for any three-variable equation). If you don’t know what I am talking about, it looks like this:

The idea behind the density triangle is you cover up the variable you want to find and then the triangle tells you what operator to use between the other two variables (multiply if the variables are next to each other, divide if the variables are stacked). Again, students see this as a shortcut that leads them to the right answer. The process or any conceptual understanding of the topic does not matter.

What do I do? After collecting these observations for the past few years, I made a radical decision this year that I have been contemplating for awhile: teach only proportional reasoning to solve unit conversions. The problem is, even proportions can be taught algorithmically (see most math classes). To deepen student understanding of what a proportion is, I have outlawed the word “per” in my class. Students often use the word “per” but have no idea what it means or implies. Instead, I have my students talk about conversion factors in terms of the relationship between the two units; for every 1 (insert unit here), there are x (insert unit here).

Since I use Modeling Instruction, my students do a lot of whiteboarding. When a group presents a unit conversion, the first questions I ask are “what were you given?”, “what unit are you trying to convert to”, “what is the relationship between those two units?” Students then set up a proportion to show that the relationship between the two units does not change.

Anecdotally, I have noticed my students talking about unit conversions differently. They talk about units in terms of relationships without me prompting them. Their work is neater and they can actually explain it instead of saying things like “I don’t really know what I did, I just multiplied the numbers and got the right answer.”  As a scientist, I know that the plural of anecdote is not data. While it is early in the year and I have not had a lot of chances to collect data, I can tell you that on their first quiz, over 90% of my students demonstrated mastery of unit conversions (I use standards-based grading). I still have work to do with the way I teach unit conversions, but the switch to proportional reasoning is the biggest step I have made in the right direction in years.