Some {Minor} Improvements on the Teaching of Limits

We just got done introducing limits in precalc.  It's a little more fun teaching it in precalculus as opposed to calculus since limits are review for my calc kids.  In precalc, my colleagues and I take a week to introduce this topic as we use a three-fold approach: understanding limits graphically, numerically, and analytically (and I would throw in verbally also).  To me, this is a fundamental concept in modern mathematics--to be able to discuss values that are either (1) tending towards infinity or (2) getting infinitesimally close to another value.

Infinity is the savior of calculus, and limits are the heartbeat of infinity.  And so, while my kids learn about limits more from an intuitive approach rather than a rigorous epsilon-delta approach, I still feel like they're doing good, valuable mathematics.  I know other calc teachers disagree with me on this one, but that's my stance.

So a few things I've done this time around that were successful (though none are my original ideas by any stretch of the imagination):

#1:  Creating Graphs
On Day 2, I had kids create all kinds graphs that satisfied certain criteria.  For example, "Sketch a graph such that the limit as x approaches 2 of f(x) is 4 but f(2) doesn't exist."  Or, "Draw a graph such that the limit as x approaches 3 from the left of f(x) is 1; the same limit from the right is -1; and f(3)=5."  The kids drew on personal whiteboards, and when I saw one I liked, I would ask the kid to put it on the Smart Board (or ask for volunteers).  Each time we had at least two examples on the board for the whole class to analyze, and I encouraged the kids who couldn't quite come up with the graph on their own to now try to make one or even replicate one that was shared by a classmate.  The good thing about having at least two graphs to look at on the Smart Board is that you can ask, "What things are the same about these graphs?  What things are different?  For the things that are the same--did your classmates HAVE to draw their graphs like that or could I change that aspect and still satisfy the given criteria?"  Really good conversations came from these graphs.  I started out pretty basic and gradually gave them harder ones.  By the end, I think every kid was able to create graphs with the given the criteria, which is exciting because my students have been somewhat unsuccessful at this in years past (because I haven't made it a big part of the learning process, which is a shame).  The last graph I had them draw was something like, "Sketch a graph such that the limit of f(x) as x approaches -2 from the right is 1; the limit as x approaches -2 from the left is 1; the limit as x approaches -2 is dne."  Of course, this is an impossible task, but it was highly amusing watching their faces as they read the question with bewilderment.  They inevitable tried to draw it, but there was a lot of erasing going on. ;)  I made sure to let them be the first ones to say something about the difficulty of the task.  Which brings me to Thing 2...

#2:  Talking about Limits

I recently attended a seminar on discourse in the mathematics classroom.  Getting kids to talk about math is something I'm passionate about and something I'm trying to get better at, so this was right down my alley, and I was able to absorb some really good information.[1]  From this seminar, there are two practices I'm trying to implement consistently:

1. Don't show approval for a correct answer nor disapproval for an incorrect answer right away.  Instead, have the kid who gave the answer explain his/her reasoning regardless of whether or not the answer is correct.
2. Don't let a kid opt out.  If all else fails, at least have the kid repeat the correct explanation of another student.

Limits turned out to be a perfect platform for me to practice both of these skills.  One of the hardest hurdles to overcome for my kids seems to be navigating removable discontinuities.  They see a hole at x=a and assume that because f(a) doesn't exist, the limit there doesn't exist either.  I can say, "a closed or open circle doesn't affect the limit" until I'm blue in the face, but that doesn't do the trick for all kids.  So, I tried to throw in a lot of practice with this concept and when a kid would say that the limit dne, I would have him explain his reasoning.  Without fail, his classmates would correct him (kindly, I might add) and boy howdy, it's so much more fun hearing explanations come out of their mouths than my own.

In one instance, I had a student that had been absent and when called upon, he was having a hard time getting to the right answer and an even harder time explaining his logic.  We soon looked at another example with a similar problem (removable discontinuity); this time he could get the right answer but still couldn't explain (but was starting to get the hint that I wasn't going to let him off the hook).  So, I had his partner explain and then immediately asked the original student to explain.  Everyone laughed as this was the third time I had asked the same question from the same kid in about a 2-minute time span, but the kid repeated what his partner said and vowed that he, along with his whole class, will now certainly answer correctly on the next test.

While that's a very simple situation and while it seems easy to implement this kind of discourse, it really isn't for me.  Yes, I love getting my kids to talk about math, but I find it takes extreme intentionality, perseverance, and patience on my part.

#3:  Limits Algebraically--Four Scenarios

The last way we learn to evaluate limits is analytically.  I start by telling the kids that we always want to begin by plugging in what x is approaching into the given expression,[2] because, ideally, our function is continuous there.  And if this is the case, we're happy and we can move on to solve the world's next problem, which surely involves limits.  This is what I ask them to write in their notes (which I'm pretty sure I learned from someone at AP Summer Institute):

[3]

My kids totally ate up the 0/0 becomes "do more work."  I mean, like literal gasps were heard in every class.  I know this is a little trick-sy, which I don't love, but I feel that once we've talked through each scenario, the kids have a fairly good grasp on the why.  Furthermore, they get a pretty firm handle on the fact that 0/0 is an indeterminant form, so they can't just assume that the limit doesn't exist...they must do more work to find out the true value of the limit.

#4:  Graph Pictionary
After we talk about limits at infinity (next week), I plan to use this activity from the Study of Change blog.  Kids get into groups of two:  one person is the "Board Partner" and the other the "Drawing Partner."  The Board Partner looks at the graph I show on the board and describes the graph using words only (hands must be folded on desk!) while the Drawing Partner, who is facing the back of the classroom, draws the graph to the best of his/her ability.  And then, we switch roles for the next graph.  The hope is that kids utilize correct mathematical vocabulary, as this will be one of the most helpful strategies to get graphs looking right.

Here are the graphs I'm using...hopefully I can hear the word limit a lot a lot a lot.

Graph Pictionary

And those are my thoughts on the teaching and learning of limits.  What are yours?  Do you have any kinds of problems that get your students talking and arguing about limits?

[1]  This alone tells me I must be getting somewhere in my professional career because typically I just leave seminars more overwhelmed than anything else and have no clue where to even begin to apply the knowledge I just received.

[2]  No piecewise functions yet.

[3]  I understand #2 is a subset of #1 but I find it helpful for students to consider the three different options of zero appearing in the numerator, denominator, or both.