Learning by understanding

I just read responses on a Linked-In forum about how to teach factoring. The answers were full of steps and technical details. Not one linked factoring to something the students knew, or gave them a reason to learn factoring.This was my response:

Before you even start factoring, make sure students have a reason to use it, that they understand WHY they're factoring. Have them graph a simple polynomial equation, like the square of (x+3), using a T-chart for values, and find the zeroes.

I think it is extremely important for the students to understand factoring in polynomials is the same in factoring, say, 96.

They need to know that polynomials are the result of multiplication, so a good way to start is to have them multiply simple things, like the results of the graph they did and other squares, and then, for example, the sum and difference of 2 terms, to see if they discover the pattern, then give them the same problem, plus some similar ones, to factor. Then move on to things like (x+1)(x+3), saving ones with a coefficient other than one for later.

Using Algebra tiles is another way to visualize what's happening, and using the "box" method, which I like for multiplication of polynomials, because it helps keep them straight, is also a good help to reverse the multiplication, which is similar to the Algebra tiles.

But if they have no clue why they are factoring, it just adds to "when will we ever use this in real life?" which is a very legitimate question. They need to know what those zeroes can be used for, too. I'm not sure all that many Algebra I teachers can carry the discussion that far.

When students understand why they're factoring, what it's used for, how the polynomials are graphed and how they came about, I think they will be much more open to the fun puzzle of untangling them as factors.

Rigor mortis or rigor percipiare

OK, the Latin in the title is my own. The second half is supposed to mean "tenacity to learn" in my version of Latin. But the title was inspired by a very thoughtful article today by Linda M. Gojak,  President of the National Council of Mathematics Teachers, called "What is all this talk about Rigor?".

Evidently people have been writing that the Common Core requirements for mathematics include the word "rigor," although she says it is not there. She and a group of math coaches investigated the meaning of the word (as in rigor mortis, but more appropriately “thoroughness”and “tenacity”) to see how it can be applied to the teaching of mathematics. They came up with the following table, which I have borrowed intact from her article.

Learning experiences that involve rigor …	Experiences that do not involve rigor …
challenge students	are more “difficult,” with no purpose (for example, adding 7ths and 15ths without a real context)
require effort and tenacity by students	require minimal effort
focus on quality (rich tasks)	focus on quantity (more pages to do)
include entry points and extensions for all students	are offered only to gifted students
are not always tidy, and can have multiple paths to possible solutions	are scripted, with a neat path to a solution
provide connections among mathematical ideas	do not connect to other mathematical ideas
contain rich mathematics that is relevant to students	contain routine procedures with little relevance
develop strategic and flexible thinking	follow a rote procedure
encourage reasoning and sense making	require memorization of rules and procedures without understanding
expect students to be actively involved in their own learning	often involve teachers doing the work while students watch

This is what teaching should be about, although I wish they'd come up with a better word, since rigor also means "rigidity" and "suffering," according to their research! That sounds more like the drill & kill methods I experienced as a student teacher, and which they define as not having rigor!

The left column should apply to all learning experiences, not just in mathematics. Children are born with curiosity, a need to be challenged and a lot of tenacity. This I experienced this past summer as my year old granddaughter tried again and again to crawl across a very difficult door opening (threshold!) until she figured it out. She was enormously proud of herself as well. I was amazed when my teacher sister-in-law got impatient with my granddaughter's efforts and just lifted her over the threshold. But the child went right back to working it out after that.

We must provide thresholds for students to cross, where they can see intriguing unknowns that awaken their curiosity. Children who are helped to everything must lose their love of a challenge and their curiosity early on. As a high school teacher I find that I have to help students regain their curiosity and encourage them through a challenge until they proudly can see they have overcome it. That is how we all learn!

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.

Doing what you love - with trash

In a recent post I wrote about a PBL Proposal I'd written for a class, where the product would be home-made musical instruments. At the very end, I included a short article about a group of people who sorted through trash as a living, who had created instruments from trash. This video shows the results. It's a teaser for a longer movie, which you can read about here: Landfill Harmonic Watching the video, I couldn't help thinking about my recent post, "Doing What You Love." These children and adults have found music that transforms their daily lives.
This also connects up with several posts about trash I've written on my other blog, Sustainable Rays, about the environment.

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!

Doing what you love

I saw this video on Facebook and had to share it. This child is completely taken up in his music. I love his expression at the end.

Compare that video with this one of some children in North Korea, who, according to the comments, are playing to provide food for their family.

Two very different kinds of motivation. I think I want my students to love what they're doing - and learning, not memorizing to get good grades for some purpose like $5 per A, or even getting into the right college.