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!

Project Based Learning to Support Math Standards

Since the one job that I've ever been dismissed from - as an Intern for my credentialing through Claremont Graduate University - questioning my ability to teach math using Project Based Learning, I just took a Teacher's Toolkit course about PBL from the UCLA Extension, Education Department. I applied for the job, and was delighted to get it, because I want math to be authentic so that students can see that they really can and will use it in their daily life. I was even promised PD on PBL, but that fizzled out soon after I started around Nov 1. After discovering that the students were drastically behind in learning what they needed of standards, I figured the best to do would be to get them up-to-date before grades were to be submitted 3 weeks later, and then use the project I'd planned, and even presented to the students orally, after the winter break.

For the Toolkit course, we read lots of articles and watched videos on sites like Edutopia and BIE, which are great sources on how to organize a project, with links to ideas for projects. Since most projects seem to use math to do its calculations, often in statistics, rather than supporting math standards,  I was looking for ideas that were specifically for math. Here are a few of the links I discovered with good ideas for projects that really support math standards.

• PBL Examples of Math Project Ideas
• Industrial Mathematics Project for High School Students - Projects
• 12 Steps to Great STEM Lessons
• Teach21 Project Based Learning

 As I read on, I decided that I'd like my final project to have something to do with the music of math, which interests me as I am also a musician, and also a physics teacher, where we touch on the production of musical tones while studying waves. I thought it would be a great way to combine various math standards in Algebra II and PreCalc with standards for waves in Physics and performance, composition and historical and ethnic instruments in Music. And there could also be some music-based readings, and the writing of song texts in ELA, and why not something about music in History as well?

Project: Building and Using Musical Instruments
Driving Question: How are musical instruments made so they can be played together harmoniously?
Concept: Students

• Use engineering skills to create musical instruments that can be played together harmoniously
• Use acquired knowledge of the math and physics of music.
• Play the instruments together in a simple composition composed by class members studying music.
• In ELA: read texts and poetry where music plays an important role, including Shakespeare, as well as song texts. Write poems that could be set to music (consider rhythm.)
• In History: discover how music influences history or history influences music
• Brainstorm what they know about music, math and science to find what they need to know.
• Are grouped according to interests, particularly which other participating subjects they are studying (math, physics, music, ELA)
• In groups will learn and use engineering principles to create a musical instrument of different types - string, wind, tuned percussion, etc. based on the knowledge of the physics and math they learn
• Learn the necessary math and physics concepts, with activities and mini-lessons using problems specific for music.
• Teach each other - through presentations, jigsawing or other means - the math, science and music they are not actually studying

Here are 2 major sources I found for this project:

• Mathematics and Music
• The Physics of Music and Musical Instruments

Interestingly, I discovered this short article about creating instruments in the magazine, The Week, a few days after I submitted my proposal. It could be and interesting addition to this project for the students to find out more about the Paraguayan project in Spanish class.

Doing the impossible is harder than I imagined!

I stopped writing here because this thing about getting a credential has become much more difficult than it was when I got the idea to do so. Neither the school I have studied at Claremont Graduate University, or EncorpsTeachers, who have also been supporting me through all of this past year with workshops, study guides, and good advice, had imagined what school administrators already suspected, that they would be hiring very few teachers. In many districts, classes are being filled up to 40 students, even in Middle School, eliminating the need to hire a new teacher, and making life difficult for both teachers and students at the same time. That means that secondary teachers have to get to know 200 students (and their families) and that classrooms built for 25 have desks squeezed in, with no space for separate activity areas, or a way to even access the walls of the classroom.

In the meantime, however, I have completed all the coursework expected of me, except for one course I'll take this Fall (in Statistics) and a concluding course next summer -- if I manage to find a job to complete the Internship training this year. I am hoping that the new government money will open up a job here or there, which may provide me a job (as well as this year's interns) and make classes a little smaller, so that it will be easier to use more creative methods for students to learn as well.

I also took a series of courses at UC Riverside Extension this summer on Science Education to supplement the Teaching Skills tests in Science (CSET) I've been taking this year to expand what I can teach.

So far I've applied for over 30 jobs this spring. I'm hoping that all the credentialed candidates have landed a position by now, as school is starting, and that schools will be more open to taking an Intern as they discover a need for just one more teacher.

Please wish me luck!

What to do when there aren't any jobs and you really want to teach...

This has been a really frustrating Fall for me. After making my decision to become a math teacher at a time when everyone said that jobs would fall into my lap, and then discovering that that is definitely not the case, I have found classes at Claremont Graduate University both inspiring and depressing - the last particularly when the speaker refers to "your students," of course. But at this point, there are still about 10 of us in secondary math, who do not yet have an Internship position.

So I have bought most of the new books available from the National Council of Teachers of Mathematics and used copies of a variety of math teaching materials, in particular Core-Plus Mathematics Contemporary Mathematics in Context from Glencoe. I've beeing reading about Sensemaking and learning through Discovery and Problem Solving, which really "make sense" to me as a way to get students interested in what they are learning. Core-Plus, which has units in Algebra, Geometry, Statistics, etc. each year, instead of separate years, looks like a fantastic way to teach math, except that it would be really hard to implement, since any student who switches schools would be lost wherever else they went. That is probably why there are so many used materials on Amazon.

I've also enjoyed the materials I discovered at the website for Geometer's Sketch Pad, which is a fun way to do geometry (and I understand other math subjects.) I found that they have great online resources for their Algebra and Geometry books, including some in Spanish. I also discovered the Prentice Hall Multilingual Handbooks - all available for almost nothing at Amazon. They included glossaries, etc. in a variety of languages, not just Spanish. But evidently Spanish is the only language that districts want to invest in. I just found a letter from a parent complaining about the creative math texts (dating back to 1996.)

I have also been observing classrooms - I'm required to observe 25 hours, including special ed and bilingual classrooms, and I've observed more than that now. What I see is teachers doing direct instruction and students doing work sheets. In some classes, text books are stacked somewhere in the classroom, or the students have a copy at home, but they are not being used. The ones I've looked at with my inexperienced eyes seems really exciting, if you want to teach by discovery and problem solving. But kids are learning how to solve problems on work sheets - and on standardized tests. I understand that there are pacing guides at the schools, that decide, for example, that next week all Algebra I teachers will be teaching solving two equations with graphing or substitution. No need to use a text book for that. No need to discover anything, when you can just do problems. Teachers tell me that the enormous textbooks just have too much material in them - and of course they're too heavy to carry around! So all the thought that went into making them (so they'd fit most any state's standards!) is just gathering dust.

To make myself more "hireable," I just did a quick (one month) review of physics and took the teaching qualification exam, CSET Physics III last Saturday. Now I have to decide whether to take Physics IV, to qualify as a physics teacher, or Science I and II (including biology, chemistry, earth and planetary science as well as physics) to be able to be a General Science teacher. Or maybe I'll just use the physics I've reviewed to provide more "authentic" problenms for my students.

Finally, I've been writing a few lesson plans, which are assignments for this semester at Claremont Graduate University - writing a total of 3 lessons that will benefit English Language Learners (ELLs) or students with other learning issues, like dyslexia or autism. I've written lessons that involve discovery and sense-making, since I'm not in a classroom trying to keep up with the pacing guide to make sure the kids do well on standardized tests. I'm afraid that my idealism will hit the dust when I finally do get into my own classroom.

After Studying Math

Studying Math
Originally uploaded by bonbayel

My recent journey through Math started innocently enough when I took Biology 101 last summer. Then I wanted to learn something about Organic Chemistry, so I bought the For Dummies book and workbook for that and a set of molecular building balls, which was lots of fun. Then my husband thought it would be interesting to look at calculus again, so we bought Calculus for Dummies and 2 workbooks. . . .

Yesterday I took the CSET in Calculus, Trig and History of Math, the last of the subject matter tests I'm planning to take to qualify as a teacher. The picture shows most of the books I've bought and devoured for this project. There aren't many For Dummies math books I haven't used!

The ones on the floor are mostly about pedagogy and classroom management, which is the next step to become a teacher. Some are required for my classes that start June. Some just looked interesting.

Last night I started a novel and signed up and played around with FaceBook. Today I rescued our beautiful Lantana bush in the corner of our patio, which had fallen down, and then downloaded pictures I took a couple of weeks ago at our family "ancestral estate" (it was a farm then,) now called Lotusland, in Montecito. The quiet before the next storm!

Doing the Impossible

Pomegranate sprouting branch,
originally uploaded by bonbayel.

I just took the California CSET Math I and II tests in Algebra and Geometry. These are the qualifying tests to be able to study as an Intern in math for California High Schools. This wouldn't be so unusual I guess if it wasn't that my last math class was in 1963 - more than 45 years ago.

In my other blog, Sustainable Rays, I wrote a short entry about words that Nelson Mandela apparently said:

It always seems impossible until it's done

I think I want to make that my theme as a teacher.

In this blog I will write my thoughts about teaching and learning. Right now it is before teaching in California schools, although I taught English and German in Danish high schools for about 14 years earlier in my career. But I think this will be an entirely different challenge.

I plan to start by observing some classes nearby, and then in June I will start my internship training. I am sure that then I will have much more to write about!