Proof by Contradiction

This post at The De Morgan Journal introduces a paper by Oleksiy Yevdokimov, on teaching math as relationships between structures. One example the author uses reminded me of my Linear Algebra students' struggles with proof and Hao's advice (in the comments) to work with them a bit on formal logic.


Do you like this example as much as I do?

I am driving a car with an 8- year-old child inside. I deliberately start naming all green cars we see around as being “red”. Red cars continue to be red ones. The child identifies the problem immediately and has a growing concern about what is happening with me. Soon we approach a street intersection with traffic lights ahead. The red light is on. I slow down and eventually stop, waiting for the green light. When the green is on, I start driving and hear a huge sigh of relief. The conclusion follows: “You are joking!”

The child’s reasoning is based on proof by contradiction: “If you are in trouble with colours, you won’t drive across the intersection when the green light is on”. 

From the child’s point of view her reasoning does not have any relation to mathematics. From the teaching point it shows that many conceptual constructions in mathematics can be successfully introduced rather sooner than later. For example, the use of proof by contradiction presented here, as well as many other useful methods and structures, can be seen everywhere through elementary and higher mathematics. The teacher’s task is to keep focus on them all the time—while moving from topic to topic, extending the content knowledge and improving problem-solving skills.

How Do You Teach Proof?

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.

I'm angry: Excel has crazy changes!

I was doing a workshop today for a few of my calculus students on using numerical integration techniques. I started to show them how to use Excel, and it had changed!

1. The columns are no longer lettered, they're numbered, just like the rows. I don't even know how to refer to a particular cell any more!
2. The formulas no longer refer to cell labels, but to the distance from the formula cell. (This is relative addressing, which was implied before, but not obvious in the formula.) 
3. The way you click to select has changed too, and I had real trouble typing in my formulas.

Disgusting!

Sonia Kovalevsky Day

Her birthday is January 15, but today gets to be her day. Cathy O'Neil (aka Mathbabe) started Sonia Kovalevsky Day at Barnard College in 2006, and it sounds like it's been going strong ever since.

Sonia is one of my heroes. Born in 1850 in Russia, she loved mathematics, fought her parents to study it, and entered into a marriage of convenience so that she could study abroad. She earned a PhD studying under Weierstrass, and still couldn't find work doing mathematics.

Eventually she got a position at the  University of Stockholm, and later won the prestigious Prix Bordin. She also had a daughter, who she raised alone. She died at age 41, when her daughter was only 12.

Mathematician, activist, and mother, Sonia was also a writer, whose novels were published to some acclaim. My kind of woman.


(Read more here and here.)

Math Movie: Inspirations Shows the Beauty of Math

by Cristóbal Vila

His explanations of the math behind this sweet movie are even more exciting, for me.


(Thanks, Murray.)

Linear Algebra: Tough Proofs for Determinants

I've been enjoying teaching this course, and I was liking the text, by David C. Lay ... until we got to chapter 3 on determinants. I worked through all the problems last weekend, verifying that everything was easy for me, and read through the proofs. Unfortunately, I thought I got it all, but I was moving too fast, and ended up in front of my class on Monday unable to really do the proofs. Embarrassing!

With my house being broken into later that day (3rd time in 5 months; yes, it's hideous, but nothing was taken this time), I never caught up this week. Yesterday and today, I've spent about 5 hours writing up the proofs for 3 theorems. I still see a few holes, but I'm pretty proud of what I've put together.

My text defines the determinant by expanding on the first row. Looking around online, that doesn't look like a standard definition, but it seems like a fine starting point. From there we want to prove that you can get the determinant by expanding on any row or column. (My text says "We omit the proof to avoid a lengthy digression." Bah! It's not math if you don't know why it's true!) My proof may still have a bit of a hole (regarding which terms are negative), but I think it's more helpful than what I found online.

My proof starts with the definition, expands completely so there are n! terms, each having n factors (which come one from each row, and simultaneously one from each column), observing the symmetry shows that we'd get the same terms no matter what row or column we expand on. The one sticky point is showing that the signs of each term stay the same. I don't think I've quite got that properly proven. Tell me what you think.


det(AB)=det(A)^.det(B). This proof is done in my text, but I felt it was done badly. I'm following his outline, but writing it up in my own words. I think there's a bit of a hole where I use L*. (The author does this step a bit differently, and I don't like his explanation.) To outline the proof:

• First, we prove it's true for any elementary matrix times a 2x2 matrix (EA),
• Then we do induction on the size of the matrix,
• Last, we show that (almost) any AB can be seen as a series of multiplications by elementary matrices (EB).

Here's my proof. What do you think? Is there a clear way to clean up the induction step?




My 3rd proof was on area of a parallelogram = absolute value of determinant (with column vectors representing adjacent sides of the parallelogram). Not particularly impressive, and I don't have the energy to do the volume proof too. Anyone want to show me a good proof of that? (We have not yet covered dot product or cross product, so it can't reference those notions.) I got this version from a mathematician I spoke with at my math circle a few days ago. I had fun using geogebra to illustrate.

Now, back to my regularly scheduled grading...

The Oakland Math Circle does Spot It

Rules of the game:

Each player gets one card. The deck goes in the middle. When you see a match between your card and the top card on the deck, you call it and collect that card, putting it on top of your pile.  Your card will always have a match with the center card, so it's just a question of who can spot their match first.

(There are 3 or 4 other games, but they're all pretty similar.)



After spending lots of time analyzing Spot It over the winter holidays, I thought it might make a good topic for the Oakland Math Circle. Two weeks ago, when I found out just a few hours ahead that I was scheduled to lead a circle, I jumped on BART, got off in downtown Berkeley, ran across the street to Games of Berkeley, bought 3 more tins of Spot It, ran back to BART, realized I'd left the tins on the counter at the store, ran back and forth another time, and made it to the circle just in time.

I had the students play a few rounds, and then we explored. The kids counted and found that most pictures appeared on 8 cards, but many appeared on only 7 cards. We eventually used Michelle's technique to figure out there were at least 57 pictures. (Pick a particular picture, the heart, for example. Pull out all the cards with a heart - there are 8 - and think about what we now know: 7 other pictures/card*8 cards + the heart = 57 pictures).

After a while, we focused on the question:  

How did the makers of this game make sure that every pair of cards has exactly one match?  

We didn't get much farther the first week. During the second week we had mostly new kids, so we started in the same place. But we got a bit farther, and tried to make our own cards with 4 pictures per card. We also had one group with a kid and an adult who had both worked on the problem before. They found a way to make every card in their deck match every other card. But their deck only had 5 cards. They figured out that they could make a similar deck of 9 cards with 8 pictures each. It wouldn't make a very satisfying Spot It game, though.

Yesterday was my third week of Spot It analysis with the Oakland Math Circle. We got back most of the kids from the first week, and played a few rounds again to get warmed up. (My first week with them, they seemed uninterested in math circle, this time they were really engaged, and much more fun to work with.) Then I let them each pick their color and gave them stacks of half-size index cards (3"x2.5"). They could choose to create cards using numbers or pictures, and were trying to make decks with 4 pictures per card, with one match between each pair of cards.

Lots of folks got the 5 card deck. We started calling that the minimal solution. I realized that was an easier solution to find than the solution that makes a bigger deck. (Although Chris and I never stumbled on it while we were creating our decks over the holidays.) One person pointed out that if they all matched on the same picture, you could have as many cards as you wanted. We called that the infinite solution. Since it would make a super-boring game, we added the condition that you have to use more than one picture for your matches, overall. People were so stuck on the minimal solution, I suggested starting with the infinite solution, and making a bunch of cards, trying to figure out when that would get you in trouble.

It turns out there are 13 cards in what I'll call the maximal solution. I realized that this brought up another question: Are there any symmetrical solutions with more than 5 cards, but less than 13?

One of my questions is whether the cards could use any number of pictures, or if there might be a constraint. (I'm thinking that the number of pictures per card might need to be 1 more than a prime, but I'm not at all sure. Yet.) Another cool aspect of this problem is that it illustrates the mathematical concept of duality. (I can't quite explain that yet, beyond saying there are 8 pictures per card, and we could have 8 of each picture in the deck.)

As I got the students into small groups yesterday, I told them, "I don't do math circles, I do math clusters." I find it's much easier to get lots of participation from the students if they're in small groups. Then my job is cross-fertilization. These last two weeks have been my favorite math circles yet. I think I'm finally getting the hang of it. My eternal thanks to Bob and Ellen Kaplan for helping me get started. I highly recommend their Summer Math Circle Institute, on July 8 to 14, in South Bend, Indiana. [Rodi Steinig, who went last summer, is doing great work in Pennsylvania, and blogs about her circles here.]

Each math circle leader has their own style, so if you're thinking about leading math circles, you need to find cool problems that work for you. My own favorites are: this problem (!), the magic pancake, playing with base 3 and base 8, and Pythagorean triples. You can find lots of math-tested-problems at the National Association of Math Circles site.

Are (K-12) Math Textbooks Getting Worse?

Dan Meyer complains eloquently that paper can't do what screens can to help students enter into a problem-solving process. It's not about the books getting worse, but about other options getting better.

Folks who want us to teach in a conventional way decry the 'reform' math texts that have been coming out for the past few decades. I generally disagree, but I've seen some folks I respect complaining, so I listen and try to learn.

I'm trying to escape from textbooks, but that takes years of experience, and lots more support than elementary teachers get these days.

Here's an entirely different take on the textbook question, from inside the publishing industry. It's a bit scary how badly schools are being used and abused in the quest for more profit. (Of course, there's also lots of money to be made by the publishers at the testing end of the schoolroom, too.)

[Edit on 2-27: Here's another post about publishers making money off education. More of a rant.]

Scary stuff.

There is so much that schools could be, but right now they are being shackled, and are not providing healthy environments for kids and their learning.

Teaching and Tutoring, It's All Good...

It's been hard to blog this school year. I've been devoting myself to my classes (and to parenting), and just haven't found time. I went to a great conference just before the semester started (Creating Balance, on math and social justice), and meant to write about it, but just didn't find the time.

I'm posting now just to say that I'm loving my classes. I feel a bond with each of the 3 groups. Two of my classes ended up being only 16 students each, and one (calc II) has 35 students. I felt guilty at first, because our classes are capped at 40 students, and I felt like I wasn't doing my share. But once the semester got under way, I was so grateful. The students in the smaller classes speak up so much more!




Pre-Calculus
Our first unit, reviewing linear equations and graphing, along with studies of circles and inequalities, showed me how much this group was going to struggle. The great thing is, they are struggling through it. The attendance is better than in my calc II class, and most of them are working hard.

The atmosphere is very lively. Students are comfortable sharing what they don't get. One student, after seeing my solution to a problem, said "I got that, but I was sure it was wrong. I did the problem 5 times, and then I threw my book across the room." I asked why she was so sure the (right) answer was wrong. Unfortunately, I don't remember what she said. I think she expected a prettier number for the answer than what she got.

We're working on triangle trigonometry right now, and a student who really struggled in the first unit is calling out answers now, and introducing the next topic by making connections (you'd think I had planted her in the audience for her great effect). Her enthusiasm is catching. I love trig, but my students haven't loved it in recent years. This time, we are so into it!

On Wednesday, I did a proof of Law of Sines for them. (-After having shown it with a numeric example on Tuesday.) They liked it. It was short enough to hold onto, and they could see the beauty in it. We all agreed that the proof of Law of Cosines was ugly in comparison - mainly because it's too long, but also because the meaning gets lost in the algebra, I think. I did improve on the textbook's version, by focusing on the Pythagorean Theorem, instead of the distance 'formula'.


Calculus II
This class meets at 8am, in a long narrow room with the students in back so far away. They don't ask many questions, and most of them are scared of looking dumb or holding the rest of the class back.  So it's harder for me to see what they're struggling with. But I walked in unprepared on Wednesday and gave my best lecture ever on Integration by Parts. Yeah, I know, lecture isn't enough. I think I got them working in groups on a problem, too.

  

On Thursday, I was tackling this one. When I got to

 

I put boxes around the integrals and labeled it as m = term + other term - 4/9m (where m stands for mess), and asked them to tell me the next step. They did. I think they saw the beauty too.

The proof (of whether my fun-for-me lecture sank in) will be in their practice. I hope they come with lots of questions on Monday.

Two students in this class asked me for extra help in understanding exponential and log functions, so we're doing a completely optional review session on that today. 10 students have committed to show up. (Now the session has happened; 7 students showed up. They were still pretty quiet. Hmm...)



Linear Algebra
I'm teaching this for the first time in over 10 years, and worked hard on prepping myself before the semester began. The text, by David Lay, is unusual, I think. In the first chapter, he introduces linear (in)dependence, span, linear transformations, and more. He also works a lot with column vectors, in ways I wasn't used to. I'm used to it now, and totally enjoying both the course and the group of students I'm working with. I just gave the first test yesterday, so we'll see how they're doing soon.

This is the first class I've ever had that's ahead of the schedule I set myself, instead of being behind schedule. I am so impressed with the students. A student I had last semester was nervous about how she did on this test, so I graded hers right away. She got an 85% and looked crestfallen. I told her I thought this might be the hardest test, and that I totally trusted that she would be able to ace the course. I think I reassured her. She and her friends work very hard, and she asks good questions.


I avoided the lower level courses this semester because I was tired of struggling with behavior issues. It looks like that was a good decision for me. I am excited and happy about my work this semester. Eventually I'll have to go back to the algebra students and try again, but for now, I'm enjoying helping students who want to learn math.


And Tutoring
My tutoring session with Artemis yesterday was a blast too. We're looking at patterns in repeating decimals. I'm learning along with him, pondering the mysteries of number theory. There was an 'extra hard' problem at the end of the chapter. It asked whether the decimal formed by the units digits of the triangular numbers (.1360518...) is rational or irrational. I said it looked too hard, and maybe we should skip it. He said, "I have a proof". (!!) He had thought (in just a few seconds) about the patterns involved in triangular numbers, and worked out why that would make this number rational. I think he'd seen a discussion of how each group of 20 numbers always adds to a multiple of ten, and that was enough to send him in the right direction. His proof looked sound to me, and it's the first time I could see how he'll eventually surpass me mathematically. His memory lets him hold so much at once, and then he gets to put those pieces together more easily.

We're getting close to the end of our text, and I've been mentioning the classes offered by Art of Problem-Solving, because I think that will be his best next step. I might take the Intermediate Number Theory Seminar with him, so he doesn't have to participate online if he's not ready.


Tomorrow is the first session of the (renamed, and slightly differently organized) Richmond Math Party. Join us at 3 if you're in the area.

Pricing Poll on the book Math from 3 to 7, by Alexander Zvonkin

A few months ago, I was able to read a draft copy of Math from Three to Seven, by Alexander Zvonkin. I loved it. But I didn't buy it, because it was $50. Now it's $42 at Amazon, but that's still too much for me.

Someone involved with the publishing of it (but not able to make pricing decisions herself) asked me if it would help if the book were a bit cheaper. The publisher will sell it in bulk at 60% of list, which is $30. I wonder if we could get them to lower the price more, if they knew how many people would buy it at a lower price.

I've set up a poll to ask how many people would buy it at $30 (which we can probably arrange somehow), and how many would buy it at $20. This price is just a pipe dream for now, but the information would be useful to my colleague, who is trying to get the AMS to understand the different market they've entered. The same problem seems to exist with the MAA. I reviewed Rediscovering Mathematics last April, which was also too expensive for my budget. University professors buy MAA and AMS books at those prices for their research, but math enthusiasts need lower prices. We are also a bigger market.

If you have any interest in Math from Three to Seven, please click here to take the survey. Thanks.


Paul Zeitz who edited the English translation of this book (originally published in Russian), said in his introduction:

As anyone who has taught or raised young children knows, mathematical education for little kids is a real mystery. What are they capable of? What should they learn first? How hard should they work? Should they even “work” at all? Should we push them, or just let them be?

There are no correct answers to these questions, and Zvonkin deals with them in classic math-circle style: He doesn’t ask and then answer a question, but shows us a problem — be it mathematical or pedagogical — and describes to us what happened. His book is a narrative about what he did, what he tried, what worked, what failed, but most important, what the kids experienced.

This book is not a guidebook. It does not purport to show you how to create precocious high achievers. It is just one person’s story about things he tried with a half-dozen young children. On the other hand, if you are interested in running a math circle, or homeschooling children, you will find this book to be an invaluable, inspiring resource. It’s not a “how to” manual as much as a “this happened” journal. ... Just about every page contains a really clever teaching idea, a cool math problem, and an inspiring and funny story.