A problem on a recent assessment I gave to my 9th graders caused me to reflect upon the role of efficiency in mathematical problem solving. In particular, how much value is there in asking students to be efficient with their approaches, if all paths lead to a similar solution? And should / could we assess efficiency?

**The scene:** this particular 9th grade class took algebra 1 in 7th grade, then geometry in 8th. As such, I find I need to embed some algebra refreshing through the semester to dust off cobwebs and set expectations for honors high school work. For this assessment, we reviewed linear functions from soup to nuts. My observation is that these students often have had slope-intercept form burned into their memory, but that the link between this and standard form is weak or non-existant. Eventually, the link between standard form and slope ( -A/B ) is developed in class, and we extend this to understanding to think about parallel and perpendicular lines. It’s often refreshing to see the class see something new in the standard form structure which they hadn’t considered before.

**The problem:** on the unit quiz, I gave a problem which asked students to find the equation of a line parallel to a given line, passing through a point. Both problem and solution are given in standard form. Here is an example of student work (actually, it’s my re-creation of their work)….

So, what’s wrong with the solution? Nothing, nothing at all.

Everything here answers the problem as stated, and there are no errors in the work. But am I worried that a student took 5 minutes to complete a problem which takes 30 seconds if standard dorm structure is understood?…just a little bit. Sharing this work with the class, many agreed that the only required “work” here is the answer…maybe just a “plug in the point” line.

My twitter friends provided some awesome feedback….

Yep, we would all prefer efficiency (maybe except Jason). Thinking that I am headed towards an important math practice here:

#### CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.

It may be unreasonable for me to expect absolute efficiency after one assessment, but let’s see what happens if I ask a similar question down the road.

Confession, I really had no idea what #CThenC was before this tweet. Some digging found the “Contemplate then Calculate” framework from Amy Lucenta and Grace Kelemanik, which at first glance seems perfect for encoruaging the appreciation for structure I was looking for here. Thanks for the share Andrew!

Yes, yes! Love this idea. The beauty of sticking to standard form in the originial problem is that it avoids all of the fraction messiness of finding the y-intercept, which is really not germane to the problem anyway. Enjoy having students share out their methods and make them their own.

What do you do to encrouage efficiency in mathematical reasoning? Share your ideas or war stories.

The danger of giving everything a name (slope/intercept, standard, the others) is that many students will not see the connections, and treat each as a separate topic.

Here’s one for them:

x(x+1) = 6

Find one solution to this equation.

Totally agree, and love your example. I try not to obsess over students doing things “my way” but I suppose we all have out built-in biases over what we think a good problem-solving approach should look like.

Thinking about “efficient” in connection with a math solution I am wondering what the criteria would be. Less paper? I am more inclined to use descriptions like “neat”, “clever’, “ingenious”, “rabbit out of a hat”, and of course “elegant”. Efficiency belongs more with computer algorithms, and is rarely an end in itself. Reverse Polish is definitely the most efficient way of representing a math function for evaluation, but it is useless for doing algebra.

It’s a sticky definition for sure, and I don’t want to blanket it by saying “less steps are better”, as sometimes in the longer journey, something unexpected is revealed. For efficiency, I’d like to see students use ideas which live higher in their tree of math ideas, and not need to resort to hammer and nails to approach each problem. You get 2 metaphors for the price of one in that last sentence…my treat.

I do suppose that this sort of scenario is why Polya’s four-steps of Problem Solving end with:

Look Back.In particular, after one grinds through computations to find a solution, it is often reasonable to ask:

Could I have derived that in another way?…Could I have seen that at a glance?Promoting this sort of self-checking has added benefits, too, such as: re-enforcing the notion that problems don’t vanish once they’ve been solved in one way (hopefully they connect to other material!) and helping to foster the sort of metacognitive thinking that is ubiquitous among strong problem solvers.