Solving Difficult Problems

Don't Panic!

My students have been brought up in the math tradition of fast is best, and there's only one way to solve problems, using tried and "true" procedures. Their previous physics teacher had selected a college textbook for them (thinking they were all AP-Physics students, which they were not, even though they were all very intelligent in their own ways.) The textbook took great pride in generating formulas for every conceivable situation, so my students had great difficulty accepting my collection of 4 formulas on the board, which I said would cover any situation they could think of that applied to Newton's 3 Laws. Time and again one of the students (usually the ones who had gotten high grades from the previous teacher) would bring that textbook to me and ask if some version of a formula was the right one to use for a particular problem. I always said, "No, use one of the four on the board." This was very difficult for them to accept until I discovered a wonderful video clip, which I show below.

I have been auditing a MOOC with Stanford Professor Keith Devlin, based on his book Introduction to Mathematical Thinking. ("Auditing" means, I'm not taking it in the allotted time, nor submitting assignments, but I'm at least watching all the video lectures.) In the very first week, he offered a video, which I called How to Solve Difficult Problems.

Techniques the Pros Use to Solve Hard Math Problems from Keith Devlin on Vimeo.

I think showing a clip from this (about half) was the turning point in getting students to understand what I was talking about.The course is intended to introduce HS students to what college math (beyond calculus) will be like, and I am auditing it, so I can mentor my students in what they need to know to succeed in college. One important thing is problem solving - not just math and physics, but everyday life away from home. So we talked about his recommendations

• Don't panic
• take your time
• take a break
• draw a picture or diagram
• write down everything you know
• learn from your mistakes, etc. 

in connection with problem solving in everyday life - deciding which college to apply for or accept, whether to buy a car, and which one, whether to date someone, etc. 
And then we applied these concepts when solving physics problems. After the video they had a much better understanding of how to solve problems. Interestingly enough, on the last test I gave them, they were very good at solving problems where they were expected to model the problem in 6 different ways, but they did poorly on multiple-choice, where I figure many rushed through in their usual manner, bringing in all their physics misconceptions.

Children in Peru write their own history on Wikipedia

I've heard about a lot of teachers who don't let their students do any research on Wikipedia.

This video shows how children in the Amazon jungles of Peru teach each other to use their little laptops, and to learn about the world through them.

They are fascinated by all the knowledge on Wikipedia - and then do their own research, photography and writing for a page about their village of Palestina, Peru.

Continuous improvement through observing scientifically

Critique and Feedback: The Story of Austin's Butterfly from Expeditionary Learning on Vimeo

This is a fantastic story about learning through observation, and making many drafts before you are through.
The class we see in the video are as fascinating to watch and listen to as the iteration of drawing they are looking at. As you can see, there are many language learners in the class - there are many Somali refugees in Maine - but there is no difference between their activity and the native Mainers.
This is from a charter school that gives children the opportunity to learn the way they learn everything else. Through repetition, self correction and help from their friends.
I'd love to teach this way.
It makes me think about the Japanese culture of "continuous improvement" which is used in quality systems in businesses, which I've read a lot about in business classes.

But teachers in Japanese classrooms also use "continuous improvement" to do "Lesson Research" while developing lessons where children learn the best - as I've been reading about in a slightly older book, the Teaching Gap, which I highly recommend! In the book, we read that education and educational methods are part of our culture, which only changes / can be changed very slowly. The Japanese evidently used their culture of "continuous improvement"from business to improve their schools. a little bit at a time.

Fall down seven times, get up eight

I just happened on a fascinating blog post, On Error, which ends with the Japanese proverb I've used as the title. I've become a substitute teacher for foreign students learning - or improving - their English on a nearby college campus, so I've been reading about teaching English to adults as "another" language (since for many students English is not just their second language!) One article I read was about how students have to learn through errors, which is exactly what that blog post is about. The article was about how to work with student's errors. Obviously, making the student look foolish or lose face would be a catastrophe, because the student would lose his desire to risk something. But if we don't correct errors, the error become "fossilized" in the student's thinking or speaking, so we are letting the student down, when he has risked making an error.

When I taught English and German in Denmark, I had a hard time convincing my students that copying someone else's translation would do them absolutely no good, because they wouldn't learn anything by doing it. A student's errors are an indication of where we have to set in do do some corrections. If you never make a mistake, it could be by chance, or because you never risk anything,
and you don't expand your experience or your knowledge.

I was using a wonderful paper-correcting system in Denmark that some smart teacher had worked out. When students made an error, I wrote a number next to it, which could be looked up in a special grammar work-book with numbered typical errors. Then the student handed back the paper with the errors listed, corrections made, and the reason explained. They learned that their errors were very effective ways to learn. But I had to grade on the returned paper, not the first one, which wouldn't be fair.

Sometimes I would also use a system of little arrows, where ↑ meant that the student had improved, and ↓ meant that she could do better (something I used more with the best students, to get them to go beyond "correct.") All of that is difficult to do with electronic grade books, unfortunately.

Oral corrections are something else again. Students want to know the right answer, and sometimes ask for correction with their tone of voice, but they don't want to look stupid (I know, that's a forbidden word in the classroom!) We can help by asking them further questions for clarification, or start the sentence for them, or ask them - or the whole class - to repeat the answer correctly (depending on the subject, of course.) Students can also work together in pairs or small groups to help each other polish off a presentation before it goes public to the whole class. That way they can "fall down seven times" gently, and stand up proudly that eighth time.

This goes for my many careers, too (which you can read about elsewhere in this blog.) I have risked much in my life, and not every attempt was successful, but I've enjoyed each time I got up again, well knowing there might be another fall - and another triumph!