In the late 90’s, at a different college, I taught Linear Algebra a few times. I wasn’t satisfied with my teaching. I could see that the students were struggling – they really couldn’t do proofs – and I had no idea how to help them.
When I decided to teach Linear this semester, I spent a lot of time beforehand studying the material. After a 12-year gap, I knew I needed to refresh my understanding. Although I still wasn’t sure exactly how I would help my students learn to prove theorems, I trusted that I’d be able to figure it out.
Just like my former students, my current students have struggled with proof. One thing that has helped is the true-false questions Lay provides in every section. I often have my students vote: “Just guess, it’s ok to guess wrong. Then we’ll discuss it.” We don’t have clickers, but they’re pretty willing to do it, and I keep finding out how much harder the material is for them than I had expected.
I use the true-false questions to make quizzes too. All my students aced the first quiz (on consistent systems and general solutions, no proofs), so I knew I had a great group. I made the second quiz a bit harder (1 question: does the span of two given vectors include a third vector given?). They still did pretty well, so on their third quiz I asked them to ‘prove or disprove’ one statement. Most of the class failed that quiz, so I gave them an alternate version the next day. They still mostly failed. I made a third version a few days later, and they finally improved.
On their first test, the ‘prove or disprove’ question had the lowest success rate of all the questions, but many of them did get it. Most of my students are used to acing their math classes, so I find myself reassuring them that this is a journey, and that I trust that they’ll get good at this before the course is over.
I think there’s one other big difference, though it’s hard to pinpoint it. The work I’ve done over the past 4 years, working on math that challenges me, and writing about mathematics, has made me more of a mathematician, and I'm sure that's helping me teach this course better. I have Bob and Ellen Kaplan, Amanda Serenevy, and a few other great teachers at the Summer Math Circle Institute to thank, along with Josh Zucker and Paul Zeitz who work with the Bay Area math circles. I also have my blog readers to thank for motivating me to keep writing here. Thank you all!
This class is the most exciting class I’ve taught in a long time. For the first time in my life, we’re ahead of the schedule I’ve set myself. That’s how good my students are. And I’m getting to talk with students about what it means to do mathematics. I think they’re learning something new, and I’m grateful to be a part of that.
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It might be good to go through a quick refresher/introduction to formal logic. I'm not convinced that this is an area of the curriculum that is well-taught, and I can see how students can breeze through non-proof based classes, having experienced only if-and-only-if statements (e.g. Pythagorean Theorem) and not if->then statements (e.g. Fermat's Little Theorem).
ReplyDeleteThis page has some nice simple examples as well as how to reframe the questions to tap into a more easily accessed social contract framework:
http://thelastpsychiatrist.com/2010/12/test_of_psychopathy.html
Good idea. I thought about it at the beginning of the semester, but didn't have any good materials handy. I like that card puzzle.
ReplyDeleteHere's a comment from Ben, who was having trouble with Blogger's anti-robot verification words. I hope people aren't giving up on commenting because of those...
ReplyDeleteIn the present context I disagree with Hao's formal logic advice. I'm
for this in the big picture of K-16 math education (I would aim for
late elementary / early middle school, and my personal choice of text
would be Raymond Smullyan's amazing puzzle books), but in a 1-semester
college class on a different topic I think it won't support the goals
of the class unless it takes up so much time it displaces those goals.
Here's my 2 cents. Not a silver bullet but I believe a necessary
condition: first of all, make a proof exercise part of the HW
regularly if you're not already doing that. Second and very important,
in all cases of proofs (on HW, tests and in class when you present a
proof or have them develop it), do everything you can to generate
legitimate uncertainty about what the outcome is going to be. The
worst thing that ever happened to the teaching of proof IMHO is that
it got disconnected from the process of students actually finding out
what the truth is. (For this reason I dig your "prove or disprove"
typed questions, but I'm saying also approach all proofs in class from
this same "not actually sure what the answer is" point of view.)
Example: recently I was working with a tutoring student on linear
algebra. The student has very little proof background. Before defining
a vector space I showed him a long list of examples of vector spaces
including the usual R^2 and R^3 but also the set of solutions to a
certain differential equation, the set of sequences with limit 0, the
set of continuous functions. Later, after developing the definition, I
gave him an exercise, "In a vector space is it always true that 0
times a vector is the zero vector? Or could it ever be something
else?" He came back with a proof (basically just a direct calculation)
that in the case where the vector space looks like column
vectors, 0v=0 always; but he didn't initially recognize the
limitation of his proof. I referred him to our list of vector spaces
to say, "okay, you have convinced me that this will always happen for
these first two items on the list, but not for the rest of the items.
What about them?" He saw that at this point he legitimately didn't
know whether 0v=0 would always be true in the other vector spaces.
This feeling of legitimate uncertainty made the next phase of our
instruction go very very smoothly. I presented a proof that referred
only to the defining properties of a vector space, and now the proof
was actually the reason for him to believe the result. Then
when I asked him a related question (can there be a second vector v
that shares with the zero vector the property that w+v=w for all w?),
he was able to come up with a good proof with very little support; and
he came up with a proof for the third question I asked by himself.
Again, not a magic bullet. The point is just that I think it's
incredibly important for learning how to create proofs, that you
actually use proofs to know what's true. The closer connected the
process of searching for proofs is to the process of finding out what
the truth is, the better.
Well, Ben, I agree with both of you.
ReplyDeleteI've seen that my students don't get why p -> q is logically the same as not q -> not p. So I'd love to spend a few minutes with puzzles that would help them get that.
I also agree with you that the more I can present the ideas of linear algebra as intellectual puzzles to solve, the better. Love your examples! Thanks.
Hi Professor Sue, Its me Calvin from Muskegon again ( the hotel clerk from 2011 and former student of yours). Just glad to see you alive and vigorous. Sorry that I have not been in touch, but sorrow grapes bring upon good wine so to speak. When it comes to the late 90's I do remember you embracing a technique that helped myself and other students quite well. I Look forward to seeing you again this year. If you cannot journey back to Michigan for the conference in August then I hope you all the best. You have always been on my mind since we last met instructor/Professor.
ReplyDeleteSincerely Calvin
Hey Calvin, Email me (suevanhattum on hotmail) so we can write each other more easily. I'll be in Muskegon in late June or early July this summer.
ReplyDeleteFor an interactive introduction to the methods of proof, may I humbly suggest my DC Proof 2.0 software available free at http://www.dcproof.com
ReplyDeleteHey Dan, is that available on a Mac?
ReplyDeleteSue, I don't have a Mac version of DC Proof, but I have run it on a friend's Mac using the Windows emulator on it.
ReplyDelete