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• Do I Dare Disturb the Universe?
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Chris Harbeck's class blog hub (innovator par excellence!)
Ryan Maksymchuk's suite of class blogs (more scribing class blogs than you can shake a stick at!)
Jim Homan's Cathoilic Morality wiki
Mr. Marti's precalculus class blog
Reversearp's (an alias I believe) precalculus class blog
Mrs. Everard's AP Calculus class blog
Image by dkuropatwa via Flickr
Every new day brings more new math (and non-math) bloggers. This is a small and woefully incomplete list. If I've failed to include your blog, or another one you know of, where the teacher has implemented the practice of having daily student authoured scribes please share it here in the comments.
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.
... 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?
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.
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:
- Construct a word problem about [this] picture that can be solved mathematically. Share your problem with a partner and solve it.
- What is the most important idea you've learned in [algebra] this week and why?
- Write a paragraph containing as many of these words as possible: ..........
- 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.
A LONG TIME COMING
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 BROAD STROKES
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.
THE TEN COMMANDMENTS FOR TEACHERS
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?"
Ten Commandments with Hebrew Numbering
Waiting for Time to Pass
The_joy_of_teaching
2 min 59 sec
Cover of Calculus Made Easy
Considering how many fools can calculate, it is surprising that it should be thought either a difficult or tedious task for any other fool to learn how to master the same tricks.
By far the best opening line for a math text ever written. Now released from copyright restrictions you can download a copy, visit the scribd.com version, or read it here.
I think I'm going to use this as the text for my High School Calculus class next year and perhaps as a supplemental text for my AP Calculus students.
Thanks to Denise for the tip. ;-)
Cool math 4 kids - math games, math puzzles, math lessons - designed for kids and fun!
http://www.primaryresources.co.uk/maths/maths.htm
maths resources for primary
Who's Counting: John Allen Paulos - ABC News
John Allen Paulos' column for ABC news: mathematics in daily life as it reads in the news.
Collection of movie clips involving Mathematics.
Visual Math Learning: A Free Online Tutorial for Teaching Math
Arcademic Skill Builders: Online Educational Video Games
Play these online or on the Wii
Math videos show how to solve problems step by step.
Integrating Google Earth in mathematics
The goal of the Place Value Game is to create the largest possible number from the digits the computer gives you. Unfortunately, the computer will give you each digit one at a time and you won't know what the next number will be. You are not allowed to rearrange any of the digits you have already placed, so think carefully before you lock a number in place! Good luck!
interesting online...play with unit cubes
This is a collection of movie clips in which Mathematics appears. I'm collecting DVDs and VHS tapes of such movies. This is a working document to be extended over time.
http://iris.peabody.vanderbilt.edu/info_briefs/k8_access_center/downloads/mnemonics_math.pdf
mnemonics instructions in mathematics
The Coordinate Plane - Math Open Reference
coordinate geometry interactive platform
interesting way to visualise math concepts
Welcome to Waldomaths - mathematical applets and videos for 11- to 19-year-olds
For the teaching and independent learning of mathematics
Online Mathematics Resources
Centre for Research in Pedagogy and Practice
The Centre brings together researchers, educators and administrators to research, develop and implement new and innovative ways of teaching and learning.
Mathematics 24x7 - There is no escape...believe it or not. Mathematics is everywhere.
Just talking about Math... What do you say?
Mathematics has a reading protocol all its own, and just as we learn to read literature, we should learn to read mathematics. Students need to learn how to read mathematics, in the same way they learn how to read a novel or a poem, listen to music, or view a painting.
Free Technology for Teachers: Math Links You Might Have Missed
"links you might have missed" series with Mathematics links
Chapter 10 - Cross Sections of a Cube
Choose a cross section, then click on the points that lie on the edges of the cube to try and create the cross section. press Create to see what cross section you made, and if you want, you can rotate the cube to take a look at it. It will give you feed
You Do The Math: Explaining Basic Concepts Behind Math Problems Improves Children's Learning
New research from Vanderbilt University has found students benefit more from being taught the concepts behind math problems rather than the exact procedures to solve the problems. The findings offer teachers new insights on how best to shape math instruction to have the greatest impact on student learning.
- It would be interesting to build a set a links to similar research results as this and discuss the implications this has for what we do as math teachers in our classrooms. - By Darren Kuropatwa
Droste Effect with Mathematica
Heavy coding but wow effect!
Professional resources - Group | Diigo
Egyptian numbering translation
Enjoy yourself while improving your maths!! Click on any title and solve the interactive exercises!!! You will find an example and exercises for each subject
Collection of Math pictures
Finite Mathematics: Everything
Can we be less prescriptive in our classrooms – and more successful with our students?
"This model for learning mathematics may be quite different from what teachers experienced themselves in the past where classrooms were less interactive, filled with little activity and conversation. Teachers were generally in control, directing all aspects of what was to be learned; different points of view and approaches seldom brought to the surface new ideas and insights and a high degree of redundancy meant that everybody learned the exact same thing at the exact same time.
Remote Access: Take Pictures With Us
Basic geometry based collaborative activity to be shared with a global audience through photos in Flickr. Activity shared by Clarence Fisher.
Posted from Diigo. The rest of Math Links group favorite links are here.
What Came Before
Part 1: Before Meeting the First Class
Part 2: The First Class
Part 3: Digital Ethics (coming soon)
Part 4: Delicious and Flickr Assignments (coming soon)
Part 5: Wiki Solution Manuals (that's this post)
Some students have already started solving and annotating their problems tonight. Feel free to watch how it unfolds on the wiki.
Over this past weekend I created a new wiki and seeded it with exam level problems. The details are posted on the front page of the wiki, but in brief, the assignment comes in two parts done over two weeks:
A Significant Contribution or Week 1
Solve a problem completely. Annotate well enough so that an interested learner can learn from you. Make the layout clean, clear, and complimentary to the articulation of the solution.
Students often think this is the hard part of the assignment. I think it's the easy part. It's more or less like a Scribe Post focusing on a single problem. Granted, some of the problems are challenging, but the timeline for this, 1 week, is supposed to make it a low pressure sort of thing. Unless they leave it for the last possible moment; most don't some do. I mean, they have a week to do something that could reasonably be assigned as an overnight homework assignment.
A Constructive Modification or Week 2
This time next week the real fun begins. (The real metacognitive work.) Each student must scan through the entire wiki looking at each solution their mates have done, find one that has an error, and fix it. They must edit someone else's work, not their own.
Week 1 was just a set up for this; this is where I think the real learning happens.
As they read through several solutions looking for errors they have to decide what is right or wrong and why. They question each line of every solution verifying each other's thinking as they read through. The hardest part for me, as the teacher, is to not say anything.
This all ties in with one of the three guiding principles I always think of as I design learning experiences for my students: Make Thinking Transparent. This is particularly well illustrated in this assignment.
Refinements or The Nuts and Bolts
The sidebar of the wiki has links to:
» the class blog.
» the grading rubric (feel free to copy it if you like, it's quite simple).
» an index of all the problems/units seeded on the wiki.
» the sitmo Google Gadget for create point & click LaTeX equations. (This makes it real easy to write math on the web.)
» equplus.net, a database of copy & paste LaTeX equations & expressions for those kids that really want to rock and roll with LaTeX.
Next in this series (when I've caught up with the backlog) is Part 6: Developing Expert Voices or Learning To Do What Mathematicians Do: create mathematics.
If you watch TED Talks you may have noticed that the visuals used by speakers at this years conference are qualitatively superior to those used in the past. TED has hired Duarte Design and assigned each speaker a small stipend to have their visuals given a makeover by the folks at Duarte. You can really see it in the visuals used in these two talks:
Barry Schwartz: The real crisis? We stopped being wise
Here's the backstory to Barry's slide makeover.
Bruce Bueno de Masquita: Three predictions on the future of Iran, and the math to back it up
Rules We Live By » Break apart big ideas into smaller bite-sized pieces.If you're new to using an Interactive White Board (IWB) in your class I think this is an excellent list to use as a starting point for slide design. When designing slides for classroom instruction these are the ideas I use as guidelines. I'm working hard at evolving how I present information to my students daily. Particularly when teaching more conceptually difficult material. This is how I introduced statistics the first time with the SMARTboard:
» Simplify the message (even when you’re talking about using game theory to predict the future!)
» Give a message space to stand out and contrast to focus attention.
» Use more visuals and less words.
» Use clear, easy-to-read charts with simple shapes and colors to add texture and clarity.
Chris Harbeck and I will be guests tomorrow morning on the Richard Cloutier Reports show at our local radio station, CJOB.
The topic is "What are our kids doing online?" but I can see us talking about adults as well as how some sites are changing the way we connect socially, personally, and professionally at all ages.
You can listen live, streamed over the internet, but we'd rather have folks participate in the chat room we've created or via twitter. We're using the hash tag #cjob.
Chris has a much more detailed post up than I do. Head over there and check it out.
Hope to see you online and on the radio. ;-) The fun starts at about 9am central time in North America.
Watch this first:
