Seeing Differently or 31 Days in October

It seems I've been talking about flickr a lot lately. Mostly in the course of discussing pedagogy and Brain Rule #10 in the context of my flickr assignment. One of the points I often bring up is the experiment I did with myself, inspired by folks who, like D'Arcy, are taking a photo a day. Every day. All year.

I just don't have that kind of stamina. I've twice in the past managed to sustain it for about a month; 31 Days. Every time the experience has changed the way I see the world around me. I notice things I've missed even though I walk past them every day; like that little tree ... I never noticed it until I started taking a picture a day for the 31 days of March, 2008.

This is what happened in October of last year when I did this again with a larger group of teachers from around the world:

Anyway, I'm starting this up again and you're invited along for the ride. I've already heard from a couple of Manitoba educators who are interested in doing this and some folks from MICDS in St. Louis might join in the fun too; they might even have a few students join us ... I really like that idea.

I'm going to tag all my photos 31DaysOct09. Feel free to use the same tag. Tagging all your photos allows you to create nifty little slideshows of them like the one I have above. Even if you use the same tag as me you can pull out just your photos if you want to create your own slideshow.

If you'd like to join, with or without your students, you can sign up for the flickr group I started. We're starting tomorrow, October 1st. If you start late, that's OK too. There are no rules. We're just going to have fun ... and maybe start seeing things differently together.

Photo Credit: Snow Came Back This Morning by flickr user dkuropatwa

The problem with math education ...

... is that there is too much emphasis on content and not enough on skills.

Math explains the world around us — makes it comprehensible — and when it's not comprehensible, when we don't understand something in the world around us, math guides our discovery ... it's all about knowing what to do when you don't know what to do.

Math is the science patterns; shouldn't we emphasize pattern recognition deliberately and explicitly in our teaching? Isn't that an important set of skills? Have I got this wrong?

Photo Credit: Sunny Side Up by flickr user code poet

Learning to Speak Math While Learning to Speak English

Multicultural Integral (sharpened)Image by dkuropatwa via Flickr

The other day I was talking to a teacher who works with EAL (English as an Additional Language) students. I was reminded of a summary of teaching tips I had assembled by scouring through a pile of research articles when my department was struggling with the issue of teaching a growing population of EAL students in our school. Unfortunately I've lost all the sources. This was originally designed to be shared with just my colleagues as a quick reference sheet.

It struck me that these tips are really good for all students.

By the way, that picture is an integral written using various languages and multilingual character sets. It makes perfect (mathematical) sense if you can decode all the numbers. Can you "read" it?

I hope other teachers find this helpful. Please add your own suggestions in the comments below.

Strategies for Teaching EAL Students Mathematics

The strategies below have been collated from a variety of resources. While all may be specifically geared towards assisting the EAL learner in math, these strategies are likely to prove effective in supporting the learning of all students.

The Language of Mathematics and Teacher’s Use of Language

"Command of mathematical language plays an important role in the development of mathematical ability"

  • Teach the language of the subject.
  • Be aware of vocabulary that has different meaning when used in mathematical contexts.
    • e.g. positive, negative, table, irrational, etc.
  • Mathematical operations signaled by several different words or phrases.
    • e.g. add, plus, sum, combine, increased by, etc.
  • Provide additional "wait time" for student responses to questions.
  • Be conscious of the vocabulary you use.
  • Simplify sentence structures and repeat sentences verbatim before trying to rephrase.
  • Rephrase idioms ("take a stab at it") or teach their meaning.
  • Clearly mark transitions during classroom activities.

Explanations and expectations need to be articulated explicitly and completely. Don't simply expect EAL students to "pick up on" assumptions, unstated premises, or subtle nuances of meaning.
  • Periodically check to ensure EAL students are understanding.

Create Language Supportive Classrooms

"Journal writing offers English Language Learners (ELLs) the opportunity to practice and develop their emerging mathematics discourse skills."

Some possible prompts for journal writing:

  1. Construct a word problem about [this] picture that can be solved mathematically. Share your problem with a partner and solve it.
  2. What is the most important idea you've learned in [algebra] this week and why?
  3. Write a paragraph containing as many of these words as possible: ..........
  4. List some things you must remember when answering this type of question or doing this type of problem.

Connect Mathematics to Students' Background and Experiences

  • Young people learn best from their own and not other peoples’ experiences.
  • Use students’ past experiences with mathematical terms to help give the terms meaning in a mathematical context.
  • The introduction of a new term should be carefully orchestrated through repetition in context and through saying it aloud and spelling it.
  • To learn mathematics successfully, many ESL students need a more multisensory approach to mathematics.
  • Relate mathematics instruction to the “out of school” life of students.
  • The implementation of “ethnomathematics” can help teachers relate mathematics to their students’ “out of school” lives.
  • Use teaching methodologies that “contextualize” the subject matter.
  • Be concerned about affective factors in the classroom.

Between the 10th and 15th year of teaching I discovered that what was needed for these children was not an emphasis on the academic but a meaningful interaction with mature adults. The relationship with a stable, mature adult is most important.

Vary Instructional Methods

"ELLs learn best when instructional methods and approaches match their individual abilities and learning styles."

  • Use a variety of methods tailored to students' needs including direct instruction, guided discovery, cooperative learning, computer assisted learning, etc.
  • Provide writing and other language development activities for EAL students.
  • Use cooperative learning strategies.
  • Encourage students to rephrase information or instructions orally.
  • Use peer tutoring.
  • Establish a "homework club".

Contextual Supports for Linguistic Development

  • Write key words on the board and other non-verbal cues, wherever possible, to present key ideas.
  • Provide written notes, summaries, instructions, and pre-reading.
  • Where possible, use the students' native language to check comprehension and clarify problems.
  • Communicate interest in students' linguistic development and set expectations.
  • Respond to students' language errors.
  • Establish a supportive environment for language learning.

In most subject areas, EAL students should be able to grasp essential concepts, if these are presented carefully, emphasized through repetition, and clearly distinguished from finer points that the students are less able to assimilate.

Assessing, Evaluating, and Reporting on Students' Progress

  • Use a diversity of measures including portfolios, observations, anecdotal records, interviews, checklists, criterion referenced tests, etc.
  • Design alternative assessment tasks including exhibits, dramatic renditions, interviews, writing samples, etc.
  • Include questions for small group discussion and individual writing.
  • Provide extra time on tests for EAL students to process the question in English, think about them in their first language, and respond in English.
  • Simplify directions in English and/or paraphrase in students' native language.
  • Permit students to use dictionaries or word lists.
  • Avoid heavy reliance on multiple-choice and true/false tests with EAL students (these involve a lot of reading and often depend on comprehension of subtle shades of meaning)

Functioning all day in a second language is exhausting and demanding. Homework can take these students two to three times longer to complete.

The Ten Commandments

From the archives ... I started writing this post in November of 2007.
A while back, Dean and Bud got me thinking. Bud's tweet contains more than a kernel of truth.

Shortly after reading Bud and Dean I was listening to David Suzuki on the radio. He mentioned in passing the idea of "in depth news reports" on television. Generally, that means they're going to talk about an issue for about two minutes. Some issues need to be explored in more depth than that. I think pedagogy is one of those issues. In particular, I think I need to explore my own teaching, articulate my own pedagogical practices, open them up to scrutiny and shore up the weak bits.

George Polya's ideas have been very influential on me and my evolution as an educator. Since reading his Ten Commandments For Teachers I have tried to model them in my practice. Although Polya (1887-1985) is no longer alive, I consider myself one of his students. This series of blog posts is my record of what I'm learning about the craft of teaching.

In chapter 14 of Polya's book, Mathematical Discovery, he talks about the teacher's attitude and structures his thinking around what he calls The Ten Commandments For Teachers. This is the first in a series of posts digging into this in depth; maybe four, maybe ten; one for each commandment. We'll see.
Although I've been thinking about it for a long time, I've got more questions than answers about these commandments. I want to share my thinking and questions here because:
(1) I want to capture my where my thinking is at today so I can come back and reconsider it in the future.
(2) I'm hoping people wiser than I might share some of their insights. I'm hoping the give and take inherent in blogging about it might push my thinking and practice; make me a better teacher.
So push back at any weak bits below or share your own teaching tip.

In chapter 14 of Mathematical Discovery Polya lists his Ten Commandments For Teachers. They have been a guiding light for me as a teacher since I first read them.
1. Be interested in your subject.
2. Know your subject.
3. Know about the ways of learning: The best way to learn anything is to discover it by yourself.
4. Try to read the faces of your students, try to see their expectations and difficulties, put yourself in their place.
5. Give them not only information, but "know-how," attitudes of mind, the habit of methodical work.
6. Let them learn guessing.
7. Let them learn proving.
8. Look out for such features of the problem at hand as may be useful in solving the problems to come — try to disclose the general pattern that lies behind the present concrete situation.
9. Do not give away your whole secret at once — let the students guess before you tell it — let them find out by themselves as much as is feasible.
10. Suggest it, do not force it down their throats.

MODELING PASSION: Be Interested In Your Subject
Polya identifies the single most powerful and infallible teaching method: "If the teacher is bored by what he is teaching it is a certainty that all his students will be too." I believe the inverse is also true: If a teacher is interested in what they are teaching their students will be too. Actually, "interest" by itself may not be enough; passion is a virus. There is a contagion in passion that can be passed on. I would rewrite Polya's first commandment as: "Be passionate about your subject." If you can't do that then I guess interest will have to do but if you don't have even that, do something else.
If you were a student in grade 8, and your teacher began a new unit of study like this what do you think your engagement with the material might be? Passion Based Learning is powerful stuff. There's something magnetic about a teacher who is passionate about what they are teaching. Passion leads to Interest leads to Life Long Learning. Kids need to see that their teachers are interested with their subject and constantly trying to get better at what they do. If we don't model the drive for excellence in what we teach why should they be interested in doing excellent work in our classes?

The teacher's most powerful pedagogical tool in their toolbox is tucked away in their attitude, demeanor and engagement with what they are teaching; we call it modeling. It's such a fundamental idea. Surprisingly, some quick searching on the net reveals very little written about the importance of modeling in education as a general pedagogical practice. Even wikipedia has very little to say about it; the closest article to this idea is called Modelling (psychology). It seems to me we need a few good educators to flesh this article out a bit on wikipedia.
A while back Dean asked, Can A Fat Man Teach Phys. Ed.? His answer: "Yes, but ..." It all comes back to the importance of modeling.

I'm reminded of Benjamin Zander who says:
Our job is to awaken possibility in others ... You know you're doing it when their eyes are shining. And if their eyes are not shinning, then we have to ask ourselves, "Who am I being that my [students] eyes are not shinning?"