What are some concrete and powerful pedagogical approaches you can leverage in a 1-to-1 or BYOD teaching environment? What sort of practices shift the focus of teaching and learning from Lower Order Thinking Skills (LOTS) to Higher Order Thinking Skills (HOTS)?



That might be a million dollar question, with many answers. Nothing works all the time, but everything works sometimes; successful learning is largely context dependent. The real question we need to ask is: "What works best in my context'?"



The slide deck below illustrates five concrete teaching practices that may be helpful in answering these questions. They grew out of several workshops I've lead with teachers discussing emerging practices in BYOD classrooms for encouraging teaching HOTS. It may be difficult to hit on all five ideas in every class you teach, and there will always be a place in our classrooms for teaching LOTS too. Nonetheless, having these rules of thumb knocking around in the back of your head can help create learning experiences for your students focused on HOTS.









Personalize Tasks to Students' Life Experiences

This is sometimes described as "personalized learning". Both the Alberta Initiative for School Improvement (AISI) and the British Columbia Education Plan discuss and share resources for personalizing learning.



Publish

When students know their work is being seen by an audience beyond the classroom it encourages them the "up their game" a bit and do better work.




Be aware that there are two sides to this coin; one positive (Social Facilitation) and one negative (Social Inhibition).



Collaborate on Group Worthy Tasks

Find ways to have your students collaborate on group worthy tasks. A group worthy task has two seemingly contradictory components: It both requires interdependence amongst students and individual accountability.

Carefully constructed group learning activities can foster students' academic and social growth and help close the achievement gap. (pdf)

Feedback

Giving effective feedback is hard. Dylan Wiliam says: "Feedback should cause thinking."




When you orchestrate feedback for your students from sources outside your classroom you also weave in the effects of many of the other rules of thumb mentioned above.



Teach Something

Teaching something is one of the most powerful ways to learn that something. As teachers, we all know that "knowing a thing" and "teaching a thing" is not the same thing. Find ways for your students to teach what  they've learned from you to others. If they do that online they are also making a contribution to the global knowledge commons.



Have you used any of these ideas in your own classroom? What was that like? Leave me a comment and let me know.

cc licensed (BY-NC) flickr photo by Jack Amick

A few years back my friend Bud Hunt published an unusual post on his blog; it was titled Going South. In just a few short sentences he shared that he'd be spending a week visiting his grandfather's garden. "As best as I can determine, the first reference on the Internet to my grandfather, a man that I know far too little about, is this one." he wrote. There was one link in the entire post; the words "this one". You can tell from the comments, not every reader followed the link.



I was talking with a couple of English Language Arts teachers today. They're planning to have their classes do most of their writing online this semester. We were talking about how they might use a Mother Blog model to do that. They'll use Google Reader to monitor the community; subscribing to both the posts and  comments of their students' blogs.



I wanted the two teachers I was talking with understand how to help their students learn to read and write hyperlinked text effectively. I shared with them the story of Bud's "Going South" blog post. It's a poignant lesson in reading and writing linked text. (In the privacy of my own thoughts: This will also be a nice memorial to Bud's granddad. In a way, he'll teach and touch the lives of generations beyond his immediate family.)



A Better Metaphor for Life Long Learning

In his book, Everything Is Miscellaneous, David Weinberger talks about the difference between writing on paper and writing hyperlinked text. Paper has physical limitations that digital text doesn't. It starts and ends. It's only so wide and so long. A paper book can only contain so much content. Even if it's one of many volumes in a larger work.



Have you ever started reading an article online, say in Wikipedia, clicked a link, then another? And another. Only to find yourself two hours later having explored a web of ideas unique to your personal interests along the way. Eventually you stop. Not because you're "finished" but because life imposes other demands on your time.



Digital text is different. It's a much better metaphor for life long learning. And you can't write linked text if you aren't reading. Lots. (It's taken me years of reading to write this blog post.)



New Media, New Process

Not too long ago Dean published a post about writing hyperlinked text. He had collected a number of comments in a storify archive and reshared a video Will had made about his writing process. (Think that through as a writing process.) Watch:





This sort of process is another thing I shared with the teachers I was talking with:

• Using a mind mapping tool, like mindmeister, that includes hyperlinks is different.


• Clipping ideas contained in text, images and video using a tool like Evernote, which allows you to create web pages & hyperlinks is different.


• Writing by stitching your ideas together from hyperlinked sources is different.


• Reading that text is different too. It matters where you publish it, online or off. Different media (paper or digital) carry different messages. 

I'm fascinated to see the student writing that emerges from this semester.



I'm also curious; is there anything different about how you teach reading and writing digital text? Any advice for us?





UPDATE 20 Feb 2012

Check out Bud's thought provoking digital writing workshop (a network of Google Docs): 

Reading 1.0 - How Digital Changes Nothing. And Everything. Is All.

cc licensed (BY-NC) flickr photo by Jack Amick

A few years back my friend Bud Hunt published an unusual post on his blog; it was titled Going South. In just a few short sentences he shared that he'd be spending a week visiting his grandfather's garden. "As best as I can determine, the first reference on the Internet to my grandfather, a man that I know far too little about, is this one." he wrote. There was one link in the entire post; the words "this one". You can tell from the comments, not every reader followed the link.



I was talking with a couple of English Language Arts teachers today. They're planning to have their classes do most of their writing online this semester. We were talking about how they might use a Mother Blog model to do that. They'll use Google Reader to monitor the community; subscribing to both the posts and  comments of their students' blogs.



I wanted the two teachers I was talking with understand how to help their students learn to read and write hyperlinked text effectively. I shared with them the story of Bud's "Going South" blog post. It's a poignant lesson in reading and writing linked text. (In the privacy of my own thoughts: This will also be a nice memorial to Bud's granddad. In a way, he'll teach and touch the lives of generations beyond his immediate family.)



A Better Metaphor for Life Long Learning

In his book, Everything Is Miscellaneous, David Weinberger talks about the difference between writing on paper and writing hyperlinked text. Paper has physical limitations that digital text doesn't. It starts and ends. It's only so wide and so long. A paper book can only contain so much content. Even if it's one of many volumes in a larger work.



Have you ever started reading an article online, say in Wikipedia, clicked a link, then another? And another. Only to find yourself two hours later having explored a web of ideas unique to your personal interests along the way. Eventually you stop. Not because you're "finished" but because life imposes other demands on your time.



Digital text is different. It's a much better metaphor for life long learning. And you can't write linked text if you aren't reading. Lots. (It's taken me years of reading to write this blog post.)



New Media, New Process

Not too long ago Dean published a post about writing hyperlinked text. He had collected a number of comments in a storify archive and reshared a video Will had made about his writing process. (Think that through as a writing process.) Watch:





This sort of process is another thing I shared with the teachers I was talking with:

• Using a mind mapping tool, like mindmeister, that includes hyperlinks is different.


• Clipping ideas contained in text, images and video using a tool like Evernote, which allows you to create web pages & hyperlinks is different.


• Writing by stitching your ideas together from hyperlinked sources is different.


• Reading that text is different too. It matters where you publish it, online or off. Different media (paper or digital) carry different messages. 

I'm fascinated to see the student writing that emerges from this semester.



I'm also curious; is there anything different about how you teach reading and writing digital text? Any advice for us?





UPDATE 20 Feb 2012

Check out Bud's thought provoking digital writing workshop (a network of Google Docs): 

Reading 1.0 - How Digital Changes Nothing. And Everything. Is All.

There's a difference between curriculum and pedagogy. Curriculum is all about what we teach. Pedagogy is about how we teach it.





There's also a difference between knowing how to do something and understanding what you're doing. In mathematics there are all kinds of "how-to", or computation skills, that kids learn and promptly forget right after the test; sometimes they forget before the test. The thing is though, it's difficult to forget something once you understand it.

  by  dkuropatwa 





A few weeks ago I was part of a panel on the Richard Cloutier Reports show on CJOB radio here in Winnipeg. There were four of us: myself, Paul Olson (President of the Manitoba Teacher's Society), Robert Craigen (Associate Professor of Mathematics, University of Manitoba) and Anna Stokke (Associate Professor of Mathematics, University of Winnipeg). Robert and Anna are one-half of the group behind the wisemath blog.





There are some things we agree on:

• All kids can and should learn basic computation skills (how to add, subtract, multiply and divide).


• It's important for kids to understand what they're doing, not just to be able to perform by rote.


• Manitoba's recent poor performance on the Pan-Canadian Assessment Programme test is not good news and we have some work to do in mathematics in Manitoba.


• We'd like to see Manitoba place at the top of future national and international tests of this sort.

  by  dkuropatwa 





Some things we disagree on. I believe:

• Learning with understanding should precede the learning of rote algorithms in mathematics.


• To say Manitoba has placed 10th out of 11 provinces and territories in the 2010 PCAP test is a gross oversimplification of the the data represented on page 24 of the report (pdf). (Those confidence intervals are important. A repeat of the same test would likely have Manitoba place somewhere between 6th and 11th place. This isn't good news, but it's a little more nuanced than "10 out of 11". People knowledgeable about mathematics should be helping the public understand these nuances and promote informed discussion.)

So the crux of our differences are two-fold:





(1) I believe Robert and Anna conflate curriculum and pedagogy and are reading the Manitoba Curriculum documents as pedagogical texts when they were never intended to be read that way. Curriculum tells us "what" to teach, not "how" to teach.





(2) Robert and Anna believe the teaching of algorithms should be student's entry point to learning the basic operations (+, -, x, ÷). I believe the algorithms should be closer to the end-game of learning the basic operations.





John Scammel blogged about his take on the views expressed on Robert and Anna's blog. John points out in the comments the clear distinction the wisemath blog draws between Mathematicians and Mathematics Educators and the populations we teach. In K-12 classrooms we teach all students. The student body in University is different. Students taking math at University want to be there. That's not true of many students in the K-12 sector; the challenges are quite different.





On further reflection, there's a third difference: public (and private) debate should be open and sidestep insult.





The wisemath site seems to reject any comments that debate the blogger's views.





What I've read in the comments on John's blog and on Anna's blog (The last sentence of the last paragraph was recently edited; it used to say all future mathematics education research has no merit as a result of the issues Anna took with the article she blogged about. I regard this edit as a positive evolution in her thinking.) seems to hold K-12 teachers in a disdainful light.





Here's the audio from the CJOB panel we sat on together. It was a 2 hour broadcast, without commercials it's about 58 min. I took out the commercials. We talked about much more than was broadcast in the moments we were "off air". That was also an interesting conversation; unfortunately we didn't capture it. Next time I'll bring along my mp3 recorder. ;-)







Download (53.2 MB)

There's a difference between curriculum and pedagogy. Curriculum is all about what we teach. Pedagogy is about how we teach it.





There's also a difference between knowing how to do something and understanding what you're doing. In mathematics there are all kinds of "how-to", or computation skills, that kids learn and promptly forget right after the test; sometimes they forget before the test. The thing is though, it's difficult to forget something once you understand it.

  by  dkuropatwa 





A few weeks ago I was part of a panel on the Richard Cloutier Reports show on CJOB radio here in Winnipeg. There were four of us: myself, Paul Olson (President of the Manitoba Teacher's Society), Robert Craigen (Associate Professor of Mathematics, University of Manitoba) and Anna Stokke (Associate Professor of Mathematics, University of Winnipeg). Robert and Anna are one-half of the group behind the wisemath blog.





There are some things we agree on:

• All kids can and should learn basic computation skills (how to add, subtract, multiply and divide).


• It's important for kids to understand what they're doing, not just to be able to perform by rote.


• Manitoba's recent poor performance on the Pan-Canadian Assessment Programme test is not good news and we have some work to do in mathematics in Manitoba.


• We'd like to see Manitoba place at the top of future national and international tests of this sort.

  by  dkuropatwa 





Some things we disagree on. I believe:

• Learning with understanding should precede the learning of rote algorithms in mathematics.


• To say Manitoba has placed 10th out of 11 provinces and territories in the 2010 PCAP test is a gross oversimplification of the the data represented on page 24 of the report (pdf). (Those confidence intervals are important. A repeat of the same test would likely have Manitoba place somewhere between 6th and 11th place. This isn't good news, but it's a little more nuanced than "10 out of 11". People knowledgeable about mathematics should be helping the public understand these nuances and promote informed discussion.)

So the crux of our differences are two-fold:





(1) I believe Robert and Anna conflate curriculum and pedagogy and are reading the Manitoba Curriculum documents as pedagogical texts when they were never intended to be read that way. Curriculum tells us "what" to teach, not "how" to teach.





(2) Robert and Anna believe the teaching of algorithms should be student's entry point to learning the basic operations (+, -, x, ÷). I believe the algorithms should be closer to the end-game of learning the basic operations.





John Scammel blogged about his take on the views expressed on Robert and Anna's blog. John points out in the comments the clear distinction the wisemath blog draws between Mathematicians and Mathematics Educators and the populations we teach. In K-12 classrooms we teach all students. The student body in University is different. Students taking math at University want to be there. That's not true of many students in the K-12 sector; the challenges are quite different.





On further reflection, there's a third difference: public (and private) debate should be open and sidestep insult.





The wisemath site seems to reject any comments that debate the blogger's views.





What I've read in the comments on John's blog and on Anna's blog (The last sentence of the last paragraph was recently edited; it used to say all future mathematics education research has no merit as a result of the issues Anna took with the article she blogged about. I regard this edit as a positive evolution in her thinking.) seems to hold K-12 teachers in a disdainful light.





Here's the audio from the CJOB panel we sat on together. It was a 2 hour broadcast, without commercials it's about 58 min. I took out the commercials. We talked about much more than was broadcast in the moments we were "off air". That was also an interesting conversation; unfortunately we didn't capture it. Next time I'll bring along my mp3 recorder. ;-)







Download (53.2 MB)

Cross posted from the November Learning blog.

---

Community First

Michael Wesch by flickr user poptech

Michael Wesch's keynote this morning was simply breathtaking. In the follow up breakout session someone asked him: "How do you stop students seeing themselves as students, and as collaborators?"



Mike sighed, put both hands on the podium and said: "That's really hard work."



He went on to explain "community first." He uses the first two weeks of class to build a sense of community and togetherness in a shared quest to solve a real world problem. A problem he himself doesn't know the answer to.



"Doing crazy things together creates community." 



Micheal plans his most passionate and enthusiastic lectures for those first two weeks. And he has his students do zany ice-breaking activities to help them get to know each other and break through the veneer of passivity they arrive in his class with. But it's not just about having fun; these activities (like human scavenger hunts) all have a serious edge to them. They have to see that they'll have fun learning here, but we are working hard at learning.



The Lesson Design Arc: schedule-research-paper-video 



The kids begin by co-creating a schedule on a wiki for the research they'll do to solve the problem they've decided to work on. They begin by digging into the problem and reading everything they can on it. Summaries of all their reading are compiled on the wiki. Typically they'll read over 90 articles, papers, or books in the first week of class as they do this. (In more typical University classes they read about three articles in the first week.) Mike guides them, having a little deeper experience in the field then they do, by suggesting other sources they might wish to explore. They continue this research and co-create a research paper for publication. When that's all done, they create very brief condensed video summaries of their research, submit them to Mike who then weaves them together into a brief (5 min?) video. All this is only possible because of the community building work they do together in the first few weeks of the course.



There's a lot more to all this, I'm just summarizing (his integrated, collaborative, calibrated peer review assessment scheme – which goes well beyond <-- that link back there – is brilliant), but that's the broad strokes takeaway I got.



When Things Go Wrong

My Pencil by flickr user jbelluch

Sometimes, when people work together closely on a real world problem things wrong. People get upset. Students goof off in class.



When that happens Mike intervenes using a ritual he learned from an African(?) tribe. It's very similar to the Talking Stick ritual used by many First Nations people of Canada. They use pencils instead. Anyone who is holding the pencil lets go of the little voice in their head that says "You can't say that." and speaks from the heart about what's upset them. The rest of the group talks with them about it. They don't put the stick down until they've resolved whatever the problem is. Mike usually goes first. Sometimes he cries while he's talking to his 400+ students. Then the next person in the group takes their turn.



A Pedagogy to Aspire To



Isn't that an amazing example of "intense imagination, motivation, emotion, and thought?" I had wanted to write about the amazing conversations going here: in the halls, in sessions, over lunch, every time someone stops me to talk really. But this morning's keynote. Just breathtaking. Good teaching is what comes from building strong relationships between teachers and students; relationships with a serious educational edge. (I hear echoes of John Seely Brown in this.)



I've got to think more about how to weave together such a set of diverse sensitivities into my teaching. How do you build a culture of caring in your class?

When I'm asked, in email or face to face, to explain how I use < web based tool > (blogs, wikis, podcasts, what have you) with my students I usually begin by saying: "That's not a short answer." While knowledge of how a particular tool works is important, it's really a distant second to the questions:

What do I have to teach?

and

How can I best help my students learn this?

This is the first in what I hope will be a series that explains how I think about answering those questions.

The slidecast and the videos are best watched in Full Screen mode.

Slidecast

Teaching (a pedagogical framework)

View more presentations from dkuropatwa.

Video

A higher quality video is available via my blip.tv channel.

Audio (8 min 4 sec)

Download 7.4 Mb

Transcript

Years of research into teaching and learning have uncovered some basic fundamental principles of how all people learn. This will just be a quick overview of how I've tried to bridge that research with my practice and what I'm calling a pedagogical framework.

But first an excursion into the world of photography. The photographic technique of framing involves finding something that draws the eye, that sits around an object to draw the viewers' attention to that thing. Here's an example.

The lady is the object of this picture the eye is naturally drawn to see her there in the centre as she's framed by the windows of the subway car.

This is a strong example of framing as we see this man walking through the arches and those arches form a frame. The eye is inexplicably drawn towards the centre of the picture where we see the man.

In this case, a more subtle frame. And yet nonetheless it's clear that the horse is the object of the picture because the tree — which we don't really look at, it kind of sits in the foreground, the eye is really drawn towards the horse — provides a frame to see the horse.

What does this have to do with teaching and learning? Just bear with me.

First the three pedagogical principles, drawn mainly from this book: How People Learn. It was written in ... I believe it was 1999 and then it was updated again in 2000. Studying years and years of research they pulled out three fundamental principles of how people learn. As an outgrowth of that book another one was written called How Students Learn, specifically in the areas of History, Mathematics, and Science. These books have been absolutely seminal in forming my thinking and providing the pedagogical framework around which I structure all the teaching that I've done.

Typically when kids come to school they think of the world as flat. They have no reason to think of it as round and they come to school and we tell them the world is round. What studies have shown is that once kids leave school and they're asked to explain what they've learned: "Well the world is round." And their conception of the world is that it's a giant pancake. That's an interesting preconception that kids bring with them when they come to the classroom and teachers need to know that. Because the first principle out of the book is that "students errors and misconceptions based on previous learning" are the first thing that teachers have to try to connect with when they're trying to teach new content.

In this example, one drawn from mathematics, kids are often taught that multiplying is repeated addition. Multiplying is not repeated addition (although that's a good place to start) and they think that the answer to any question, when they multiply, has to be bigger than any of the numbers they started off with. Given a problem like this, fractions, well kids find that really hard, they just mul... they see the two the three and they know they have to have an answer that's bigger than either of them. If we know that that's the preconception that kids bring to the classroom then that can inform our teaching in constructive ways.

The second principle out of the book is that understanding requires not only factual knowledge, knowledge of basic facts, but an understanding of the basic ideas or big ideas of the discipline, whatever the discipline is that happens to be that you're teaching. Because knowledge isn't actually built in hierarchies knowledge is actually built in networks. For example, take something simple like three quarters. It seems like a pretty simple idea. But the number three quarters can be represented in a variety of ways. All of these are equivalent ways to write three quarters. It might have meant money, seventy five percent, it could be the ratio of three to four, point seven five or another way to write the fraction three quarters. Now even this extends beyond that because, "three quarters", well I could have been saying the money, three coins, three twenty five cent coins, which is seventy five cents. You see all these ideas are connected one to the other.

I might have been talking about a piece of cake; that I have three quarters of a cake. And it's implicit in that idea that each of those quarters is the same size. That each piece of cake has to be equivalent in size. But that's not always true with all ratios.

For example the ration of three red Smarties to four blue Smarties. I've got seven Smarties in total. And those Smarties don't all have to be the same size for my ratio of three red to four blue to be true.

These implicit assumptions make learning this material difficult for kids, and we as teachers make assumptions because we understand from the context that these things are clear. It could be seventy five percent off everything. That these ... All of these ideas actually live a network all one related to the other. The underlying assumptions that we make as experienced learners is that we take, from the surrounding context, what the meaning of each of these numbers is. And yet each one is related to the other.

That network has to be made explicit. That network of concepts has to made explicit to students. No matter what fundamental idea you're trying to get across to the students. In this case, the idea, the big idea, is really one of proportion.

The third principle out of the book is that "learning is facilitated through the use of meta-cognitive strategies". The degree to which we can get kids to think about what they're learning as they're learning it will deepen that learning. And they found that this made only moderate increases for high performing students, but for low performing students the use of meta-cognitive strategies made for dramatic increases in their performance.

Error analysis is a great example of this kind of thing; you've probably caught that one really quick. But as you look at this picture and try to find the error look how you're thinking about it. Look how you're paying attention to certain details; finding what's the same, what looks exactly identical? Where is the difference? Where is the thing that stands out one different from the other? And if you pause to reflect even as you're thinking about this now you'll notice that you've deepened your own thought as you look for the error in just something simple, like a picture of three Mounties.

So that's the framework. That's the framing. That's the ... Those are the ideas that sit around any pedagogical approach that you want to take in class. Any time you want to structure or design a learning experience for students these three principles:

Connect with kids preconceptions.

Learning should be networked. Ideas are networked. You need to understand basic facts but also the big ideas around which knowledge is structured.

And engaging kids in meta-cognition as they're learning.

This provides the framework for how we teach.

There's one last thing not to exclude in any of this and that's "community". It's alluded to in that book but not listed as one of the fundamental principles; but "community" is a pretty big deal. Because the degree to which we can get kids working with each other and collaborating and helping each other in their learning, which is what genuine learning looks like, is the degree to which they can deepen and accelerate their own learning.

This provides a wonderful example of exactly what I'm talking about: Why do geese fly in this "V" shape? It seems quite distinctive to see the Canadian geese flying in that pattern. It happens in the Fall, around October, and then again in the month of March as when ... as the geese leave in October and return in March. Why the "V" pattern? Because the flapping of the wings of the lead goose actually provides a little lift for the geese just behind them. And that's true for each goose behind every other goose. And so they're able to fly greater distances. Of course this puts undue pressure on the lead goose. So throughout the flight the geese are constantly shifting positions and rotating their spot in the "V" shape flight pattern. So that different geese take the lead at different times. And by working together the geese are able to fly for much greater distances than any of them would be able to fly on their own. And together they accomplish great things which is the very distant migration patterns of the Canada geese. Of course over short distances, if everybody goes their own way, well, yeah, you could have some success, and when they all land that's what it looks like. And they all kind of come down and each one chooses their own pattern and the "V" is broken up. And occasionally, you know, you need to do that, but by and large, for the most part, it's through that collaboration that great things become possible. And that's part of that framework that I talked about.

It's just like we learned in kindergarten: When you go out into the world, hold hands and stick together.

... was the title of the Keynote address I gave to 500+ educators at this year's B.Y.T.E. conference. The sense I had of the audience was that many of the ideas I talked about here were new to them. A quick survey of the room while giving the talk revealed only about three people had heard of Wolfram Alpha and I think about the same number had heard of the TPACK Framework.




I hope I get a chance to do this talk again. There are a number of things I think I could have done better.




Anyway, if you're interested, here it is in various formats. You pick how you'd like to take it in.




Audio





Download (20 MB)




Slidecast







Ustream Video (thanks to Chris Harbeck

actually, it was 11 minutes and 45 seconds.

This is the slidecast from one of three talks I gave on Friday October 9, 2009. I was in Virden Manitoba participating in the NIBBLE Conference for the Fort La Bosse School Division.

More coming soon. Everything will be aggregated on the Senior Years Information and Communication Technology wiki I'm maintaining to share all the work I do in Manitoba with teachers across the province ... and you.

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.

From the archives ... I started writing this post in November of 2007.


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?"

Cross posted from a ning community I'm in.

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Watch this first:

OK, so this is going to sound weird coming from a math teacher — I'm liable to be run out of the club for saying it — but, in most subjects, does every kid have to learn exactly the same stuff?

Here's what I'm thinking (and I don't think it's an original thought):

Don't send them home to read and listen to the lecture, send them home to take in a short (10-15 min video) or even a micro lecture. Then change the classroom into more of a lab or studio environment. Each kid produces a paper or other artifact of what they've learned and shares it with the rest of the class either face-to-face or online; they become expert in the area they've chosen to explore and at the same time develop the research skills to learn related content when needed.

The question I try to ask myself is: What is the value added for my students by being in the same room with me? If I recorded my lecture (video or audio) and they watched it at home, did the assignments and handed them in, would they be missing something by not being here physically?

I do think my students gain value by being in the same room with me, but most often when I speak very little. I let them work through the problem(s), debate and defend their work with each other, and only towards the end, when they've collectively sucked the marrow from the bones of the problem do I either ask another question that fires them all up again or draw their attention to the finer points of how best to share their thinking on paper.

This is what the video I embedded above suggests to me. I know I'm not there yet, but man! I'd like to be.

I've been using a SMARTboard in my class for a little more than two years now. I know I've seen dramatic growth in the way I teach and present information to my students in no small part because of it.

I'd like to publish a series of posts transparently sharing how I use the SMARTboard to teach math and encourage anyone who reads/hears it to leave me suggetsions on how I might better use the affordances offered by the SB. Please share your critiques, comments, concerns, questions, complaints, confusions, uncertainties, anxieties, and suggestions for improvement with me by leaving a comment on this post or directly in the VoiceThread I've created to share my process.

I'd encourage other teachers to do the same; we could learn a lot from each other this way.

LINKS FOR THIS LESSON

Teaching Slides


Student Authoured Scribe Post

Thanks!

Why YouTube?



In the comments to my last post I was asked why I chose to use YouTube instead of TeacherTube as a platform for my students to share their videos. I answered:

I chose YouTube for three reasons:

(1) That's where the kids go for video. They feel comfortable there and have their own accounts from which they publish video outside of school. This allows me to model some digital ethics vis a vis what do you and what don't you publish on YouTube.

(2) That's where all the people are. There is a significant math community on YouTube and I am hoping to draw their attention and perhaps participation which would add value to the assignment in terms of authenticity and audience.

(3) Kids think TeacherTube is for teachers, not them. That detracts somewhat from the value added by publishing publicly online.

A number of people have left comments for my students on their class blog. Thank you so much for that. In one of those comments pirategirl mentions how slow TeacherTube can be. I guess we can add that to my list of reasons above.

Copyright


As part of this assignment we talked about copyright and I emphasized that using copyrighted material (music) is considered stealing unless they have permission to use the content. Generally, I believe that using 30 seconds or less of copyrighted music falls under Canadian Fair Dealing guidelines. (If I'm wrong about this I welcome the correction.) As part of this discussion I pointed the students to Jamendo where they could find free music to use without worrying about any copyright infringements.

I'm really proud of the way they all dealt with copyright issues and how they credited all the sources they used. Click through to their YouTube videos (see below) to read the details they included not only in the videos but in the informational text that accompanies their work.

The day before the deadline for students to publish their videos one of them left this comment on their class blog:

Maybe JAMENDO should be my name =D

it helps a lot.. thanks..

STATUS: currently cutting down 1:30 to 0:30 s.... 0__o

I was particularly chuffed about that "status update." (See the short podcast below.)

The Videos & The Podcast


Here are the four videos that the student groups generated (the 30 second time limit did not include credits) followed by a brief podcast reflection we did on the last day of classes before the winter break. Give it a listen, this project isn't over yet ...

Ben and Zeph: Calculus Commercial

Kristina, Jamie, & Joyce: YE OLDE DERIVATIVE

Paul, Shelley, & Yinan: Team PSY Derivatives Commercial

Francis, Lawrence, & Justice: Calculus commercial complete

The Podcast Reflection

(Download mp3 file, 3.3 MB)

One of the things we learned was that 30 seconds is too short. The next round of videos will have a 60 second time limit. They'll all be published by the end of January. Stay tuned. ;-)