Learning Goals in Math

I had a very interesting conversation with my friends Janice and Celiza the other night.  Janice and Celiza are currently Curriculum Consultants in two other school boards in Ontario.

We were discussing the use of Learning Goals in Math.  Janice was of the mind that we shouldn't be naming the Learning Goals up front at the beginning of the Math lesson if we are truly teaching through problem solving.  I understand her point.  If you are teaching Math through problem-solving - which I hope you are - then you are using a constructivist approach, and you want the children to construct their own learning.  You don't want to give away the punch line.  Janice was adamant that the Learning Goal should come out during the Consolidation (the third part of the three part Math lesson).

<For a brief review of the Three-Part Math Lesson click HERE.>

I was very fortunate to go and hear Dr. Marian Small speak last May (I reference her often, I am quite a fan of her work).  She addressed this very question.  She said that we have to tell the students something about what we want them to do.  She asked (and I quote) "Can we say: 'We are going to develop strategies to compare two fractions'?"

What I liked about Marian's talk was that she never told us what we should be doing.  She only ASKED us what we might want to consider.  The fact is, there is no right answer here.  It is trial and error.

I have tried it both ways, not telling my students anything, and also giving them a simple statement about what we are hoping to learn about.  Not what we are going to learn specifically, but what we are learning about.  I have found that giving them a Learning Goal up front helps give them more direction.  It sets a purpose up front for why I am giving them the task that I am giving them.  And, I admit, it speeds up the problem-solving and consolidation.  The fact is, I don't have unlimited amount of time to teach them everything I want to teach them.  I only have 9 months to teach them 10 months of curriculum (because I have EQAO testing in the beginning of June).

What I don't do is give them Success Criteria for the content expectations up front.  That is what I want them to come up with in the consolidation.  I DO however, give them Success Criteria for the process expectations up front. (The process expectations include communication, representing, selecting tools and computational strategies, problem-solving, reflecting, reasoning and proving). Confused yet?

For example, I might write:

• We can reflect on the reasonableness of a solution 
• If we determine our solution doesn't make sense we can look for a different solution
• We can share our solution with others in a way that makes sense to them
• We can explain our thinking so that someone else "gets it"
• We can connect our solution to someone else's solution
• We can restate the problem in our own words

Here was our Math Learning Goal from Thursday and Friday:

For our Minds On, I gave the students fractions to add to a number line.  This allowed me to differentiate by giving different students different fractions.  As the students placed their fractions on the line, I asked them how they knew where to put it.  "Tell me what you are thinking in your head as you are deciding where to put that fraction."

Here are some strategies we came up with for comparing fractions in our Consolidation:

We are still using the "ShowMe" app and "Explain Everything" app to show how we solve a Math problem.
Here is a video from one of the problems my students solved on Friday.  I was only going to give them one problem, but some of the students solved it so quickly and easily, we went to the text book to find a second problem.


I think it is important at this point to mention the obvious.  This lesson could not have happened if it weren't for the previous lessons.  We spent Monday, Tuesday and Wednesday reviewing what fractions ARE.  Before students could begin comparing fractions, they had to fully understand what a numerator and denominator are, and they had to understand what mixed and improper fractions are.  For example, many students knew that an improper fraction was a fraction with a numerator larger than the denominator, but very few students knew that this meant we were talking about a number greater than one.

I also had one student who could compare fractions by using a common denominator.  He had learned how to do this in a previous school in grade five.  But he did not know how to represent those two fractions, and he did not know how to compare those two fractions without using a common denominator.  For example:  students should know that 5/6 is less than 7/8 because sixths are larger pieces than eighths. Just because they can convert them into 20/24 and 21/24 doesn't mean that they understand that concept.

I guess my point here is this:  You have to be extremely deliberate in everything you do when you are teaching.  Before you post a Learning Goal, you have to think, "How should I word this?" "How will knowing this help my students?" "What do my students already know and think?" "What skills do they need?" We have to be reflective practitioners; I am very fortunate to have friends that are interested in having these conversations with me!

The Language Block

One of the most common questions I get asked by other teachers is "What does your Language block look like?"

I have a really difficult time answering this question; it is something that would really take days to describe.

I think the problem lies in the fact that we are so used to thinking in terms of our "timetable".  By definition, in our timetable we show each subject and strand as a discrete entity.  But that is not the reality at all, at least not in my class.  

For example, last week we were watching videos about the new pope, Pope Francis.  We watched and discussed a video on YouTube about how to become a pope, we watched a CNN video about Pope Francis' first public appearance, and we read a couple of related passages from the Bible together, John 1:41-42 and 21:15-17.  At the end of the period, one of my students said, "Was that Religion or Language?"  I answered, "Yes."

Admittedly, I have "Language" penciled into my timetable, along with "Social Studies" and "Science".  I have the Language block further delineated into "Reading", "Writing", and “Word Study".  But in reality, any mixture of those items is happening simultaneously. I have tried to create a timetable that accurately reflects what happens on a day to day basis in the classroom, but it isn't a true picture of what goes on during the day. 

For me, what is truly important is ensuring that I have a comprehensive Literacy program, and by "comprehensive" I mean all-inclusive. I make sure that my students are reading and writing something every day.  I also want to make sure that there is a balance of Modeled, Shared, and Guided Instruction along with oodles of time for Independent practice.  

How do I build it all in?

Well, I spend a great deal of time during the beginning of the school year teaching my students to be independent, self-directed learners.  I do a lot of explicit instruction on how to become exactly that, and then I help them to build their stamina gradually.  So my Language block looks vastly different in the fall than it does in the spring.  In the fall, I start with a modeled or shared reading lesson, then they read for about 10 - 15 minutes independently while I take students for guided small group instruction.  I provide some direct instruction on Writing, then they work on a writing piece for 15- 20 minutes independently; I intersperse direct instruction in Reading, Writing and Word Study daily with independent practice. I decide when it is time to switch activities.  

But that is not what my Language block looks like at all at this time of the year.

I'm at that happy place right now where my students can work for 60 minutes independently (and by "independently" they may be working by themselves or collaboratively with a partner), without being off-task or disruptive (with the exception of one or two students who need occasional reminders and re-direction). 

In order for this to be possible, the students have to have highly engaging tasks that they are motivated to do, and that serve authentic purposes.  My Language block truly becomes a Language Workshop.  I list all of the items I would like students to accomplish by the end of the week, much like a menu. Students then select from the menu what they would like to work on.  Sometimes they choose based on what they enjoy doing the most, other times they choose based on what is a priority, or what they are struggling with and need to spend more time on.  I prefer to start with a mini-lesson on Reading, Writing or Word Study, but lately, I have an EA at the beginning of my Language block, so I take advantage of the extra adult in the room, and we begin our Workshop right away.  That way, she can be that support person in the room for students while I take groups to the horseshoe table for small group instruction.  I save the last 15-20 minutes of the Language block for some direct whole-class instruction

Here are a list of activities my students currently have to work on:

• Become familiar with word patterns in Word Wall Words by writing sentences and paragraphs using those words
• Post to their blogs
• Create or add to a Discussion Thread about the novel I am currently reading aloud to them
• Revise their recounts
• Swap their recount with another person in the classroom, and using Co-created Success Criteria, provide one another with descriptive feedback
• revise their recount based on descriptive feedback given by a peer
• Independent reading - focus on being metacognitive while reading

On Friday, I asked my students if they enjoyed working this way.  They gave me a resounding 'yes'.  When I asked why, they said:

·         Because except for the Word Study work, everything we have to do has a purpose
·         Nobody is fooling around and everyone knows what they have to do
·         The work is not too hard

·         The work is fun, especially when we get to use electronic devices

·         Nobody bugs anybody else when we work like this

Is my Language block ALWAYS like that?  No.  It depends on what we are working on.  When we are learning a new writing form, for example, I have students  in collaborative groups working with writing exemplars so that we can co-create success criteria for that writing form.  I find that whole process takes up 80 minutes. Next week, I'll explain what my Language block looks like when we are working on an Inquiry project, because that looks different again!  Stay tuned...

Digital Literacy, Web Literacy, Media Literacy, Information Literacy - or Just Plain LITERACY???

We are well-into our third topic of #etmooc.  We were asked to consider the following questions:

• What does it mean to be digitally literate? 
• What is the difference between being digital literate and web literate?
• What digital competencies and skills do your learners demonstrate through their daily use of technology?

I can honestly say, that until this #etmooc, I never considered these questions before.  I thought of digital literacy as being equivalent to the ability to use Web 2.0 tools. But  I never even stopped to think what is meant by "Web 2.0 tools". 

So, today, I looked up "Web 2.0 tools" and my first hit brought me to Discovery Education which said "Web 2.0 is about revolutionary new ways of creating, collaborating, editing and sharing user-generated content online. It's also about ease of use." 

As I've pondered these questions this week, I became aware of how very little I have considered this topic in the past and how little I know about it.  

On Wikipedia, I found the following definition of digital literacy:

Digital literacy is the ability to effectively and critically navigate, evaluate and create information using a range of digital technologies. It requires one "to recognize and use that power, to manipulate and transform digital media, to distribute pervasively, and to easily adapt them to new forms".^[1] Digital literacy does not replace traditional forms of literacy, it builds upon the foundation of traditional forms of literacy.^[2] Digital literacy is the marrying of the two terms digital and literacy, however, it is much more than a combination of the two terms. Digital information is a symbolic representation of data, and literacy refers to the ability to read for knowledge, write coherently, and think critically about the written word.

In our province, we report on four strands of Language - Oral, Reading, Writing and Media Literacy.

I'm not even sure what is meant by "Media Literacy".  Is that the same as "Digital Literacy?" It seems more encompassing than Digital Literacy.  In fact, isn't every form of communication a medium and therefore part of Media Literacy?  Or should we consider Media Literacy to refer to multi-media literacy?

You see, the more I think about it, the more questions I have and the more confused I get!

So, today I decided it doesn't matter.  

What is important is that I consider what it means to be LITERATE.  To me, being literate means being able to use expressive and receptive communication skills to collaborate, innovate, create, and participate with others.  Being literate means being able to think critically about incoming information and being able to synthesize that information.  It means being able to share ideas in a coherent way so as to be understood by others.

When I was a Girl Guide, (a long, long time ago), I had to learn how to use flag semaphore, (a way to communicate using flags) in order to earn certain badges.  I never understood why I needed to learn flag semaphore.  What good is knowing a form of communication if there is no one out there to communicate with using that particular medium?  I don't know how to speak Latin, and yet I still consider myself literate, but fifty years ago, I might not have been considered literate without a working knowledge of Latin.  Years ago, you had to be fluent in Morse code to be a pilot - that is not longer the case either. 

What it means to be literate changes with time.  As the world changes, and we develop new ways of communicating with one another, we must adopt these new methods to be currently literate.  What it means to be "literate" changes as our modes of communication change.  I believe that in this 21st Century, in order to be literate, we have to have a working knowledge of digital tools in order to communicate most effectively.  And that is what Digital Literacy means to me. 

Networking in My Zone of Proximal Development

CC licensed photo  shared by Flickr user edenpictures

When my nephew Jack was five years old, he was afraid to swim in the deep end.  My sister, Jennifer, put him in swimming lessons but Jack politely refused to go into the deep end of the pool with his swimming teacher.   I had a pool in my backyard and Jen asked if she could bring Jack over during the summer to practise.   I suggested she also bring a friend for Jack, someone who could swim a bit better than himself.  

I had noticed over the years that my own children didn't try to copy accomplished swimmers when they were learning to swim, but if they saw a peer, someone who was just a bit better at swimming than they, trying out new things in the water, they would watch carefully, and then attempt it themselves.  

This observation proved true for Jack as well.  He came swimming with his little friend Natalie.  Natalie was a novice swimmer as well, but she was just a bit more daring.  As she dog-paddled across the width of the pool in the deep end, Jack studied her, and it wasn't long before he was doing it too.  The next thing we knew, he and Natalie were challenging one another to jump in, or to swim the length of the pool; they even made a "slide" out of a float and slid into the deep end laughing and screaming. 

My etmooc experience has been very similar to Jack's swimming journey.  I have enjoyed the webinars greatly, but sometimes, the information is over my head and I have to watch the archived videos again and again to try to understand.  (It is driving my husband nuts!).  I have also been following most of the Google+ Etmooc Community posts, but many of my fellow etmoocers are experienced tech users with professions in IT and much of what they share is beyond my scope.   

Just like Jack, the people I learn most from are my peers, the etmoocers who know just a bit more than I do.  I think this is why the networking component of the etmooc is so critical.  I also think this is why I have been enjoying and learning so much in the etmooc.   Reading a range of all of the posts, I have found that I understand some posts better than others - those are the people I follow and network with, we learn together that way.  

Vygostsky's Zone of Proximal Development suggests that I learn best when a task is just slightly out of my reach and I am supported in learning how to accomplish that task (in school, we call this "guided instruction").  Sometimes, an expert isn't the best person to support me in that learning, they simply know too much, and perhaps forget how little I know.   But if I learn with someone within my ZPD, someone who knows just a bit more than I do, we can support one another to accomplish things we couldn't do alone.

This video is a quick refresher on ZPD:

The same holds true in my classroom.  I need to talk less and let my students talk more because it is through the conversations they have with one another that most of their learning will take place. 

A Digital Math Story

On Monday, at a 21st Century workshop, I learned about the SAMR, a model for the use of technology in the classroom.   But even before I saw this model, I had already decided that if I am going to be teaching with technology, I want to use the technology to teach differently.  I don't want to use technology as a substitute for our old Blackline Masters.  I started teaching with the SMART board five years ago, I want to do more with it than use it as a fancy whiteboard.

Since I started the etMOOC (Massive Online Open Course about educational technology), I've been  contemplating how to make students creators of knowledge.  21st Century learners are collaborators, creators, and sharers.  I knew that before, but the etMOOC has been helping me to make that into a reality in my classroom. 

During that same Monday workshop, I was introduced to a new version of Bloom's Taxonomy made for iPads: 

The point, of course, is that the highest level of thinking is creating or synthesizing, so I have been spending my week exploring how I can have my students create their own knowledge. 

I liked the way +Michael Buist uploaded a video of his students' completing a math task.  You can see in the video how he scaffolds their learning and understanding.  I was truly inspired by that video.  To me, this was a Digital Math Story.

I decided to have my students create videos of their math solutions in a similar way.  I used the ShowMe app because it is free and so easy to learn and use.  My students quickly caught on and in their collaborative groups, they used the ShowMe app to explain how they used the volume of rectangular prisms to solve math problems.  As always, I was walking around the class while the students were doing their work.  I was thrilled because they all seemed to be understanding the math concepts.  I asked them questions, they gave appropriate answers.  During the consolidation of our math class, we watched one of the videos.  We made comments and asked more questions.  Everything seemed great.  But then when I got home, I watched the rest of the videos, and I discovered so many misconceptions as I listened to the conversations they were having while they were solving the problem.  It was incredible learning for me.  

The next day we watched the videos and discussed everyone's misconceptions.  We also reviewed the first part of the 4-Part Problem Solving Model, because I realized that while my students understood the math, they weren't always understanding or answering the question.  

Today, we solved another math problem; this one required that they use their knowledge and understanding of the surface area of rectangular prisms.   I asked only two groups to create videos.  I chose one of the groups because they had previously demonstrated that they made certain to understand the problem before moving on.  Then I showed their video to the class as a model for how to go through the first step of the Problem Solving model.  

One of my students posted this on Twitter.  I asked her to explain how making the ShowMe videos was helpful and she said, "When I explain it, I understand it better afterward".  And THAT is why it is so important to be a knowledge creator, collaborator and sharer in the 21st Century!

Making A GIF- If at First You Don't Succeed, Try, Try Again - And Then Ask Your PLN for HELP!

This is my very first (and only) GIF.  I cannot begin to explain all of the learning that went into my being able to create this little baby - it's the story of my backyard by the way - but the hoops I had to jump through to create this animated image are actually irrelevant.

What IS important, is that when I first began watching Jim Groom's session on creating GIFs, my immediate thought was, "This is over my head, not for me, it will be a long time, if ever, before I can do something like that!".

But all week long, people in our etmooc community were posting their GIFs.  How hard could it be?  I felt out of the loop.

With the help of my husband, (much against his wishes, he has been learning right along with me!), I managed to create a GIF using the free GIMP software suggested in the Tumblr tutorial we were sent in our etmooc Digital Storytelling introduction.

I was so proud of myself for being able to create this silly little GIF; but then couldn't figure out how to upload it to my blog or share it on Google+.  Each time I tried, it would lose its animation.  So I posted a question in the Google+ community.  With the help of +Alison Seaman and +Andrew Forgrave I was finally able to upload it.

What did I learn?  I learned that I can do things I never thought possible; that an online PLN is an incredibly powerful resource; that learning is not easy, but persistence and determination will be rewarded; and that success feels really, really good!

As I re-read this post, I realize that as a student, learning was always easy for me.  Math, Language, Science, loved them all... I never had trouble in school.  Going through this etmooc, I feel, for the very first time, like a struggling student.  It has given me so much empathy for those children whom I serve that just don't "get it" the first time around.  I think that is the most powerful piece of learning in this etmooc for me.

Digital Storytelling - What Is It?

I thought I was on information over-load after the first three weeks of the etMOOC (Massive Open Online Course) I signed up for.  Last weekend, we were introduced to our new topic: Digital Storytelling.  I have been to workshops before on Digital Storytelling, actually even created some of my own "stories" using Movie Maker, so I thought this week would be easier.

I was wrong!

It was suggested that we start the week by considering the definition of Digital Storytelling as found in Wikipedia:  "Digital Storytelling" is a relatively new term which describes the new practice of ordinary people who use digital tools to tell their 'story'".  I soon realized that there are as many different definitions of Digital Storytelling out there as there are computers!  Everyone has a different idea of what Digital Storytelling is.

My children say I process slowly; actually, their term is "Mom's buffering".  Well, I have spent most of this week "buffering".  While I've been reading, viewing, and listening to all of my fellow etmoocers' different submissions, their "stories", I've been trying to decide what "story" means to me.  I believe the term "digital" is actually irrelevant - what interests me is what do we mean by the word story.

This buffering time made me realize that without consciously thinking about it, I had come to see the word "story" as being synonymous with the word "narrative".  I  no longer believe that to be true.  I believe now that a narrative is just one form of "story".  How narrow is our thinking so often in Education!  In our school board, we teach the children that there are six forms of writing:  Narrative, Persuasive, Report, Recount, Explanation, and Procedural.  Isn't that just so silly?  Why do we always try to pigeon-whole things in Education?  Isn't almost EVERY good piece of text persuasive?

And so what are my blog posts?  Are they recounts?  No - I don't think so.  Are they informational?  Certainly they contain bits of information, but they are definitely not reports or explanations.  I think of them as reflections.  What form of writing does THAT fit into?

And so I have discovered that I fell into the educational trap of thinking in black and white, thinking in the box, and was shocked to realize that at some point in my adult years, I had narrowly limited the story to a narrative!

STORY is so much more!  I have been trying all week to put into words what STORY is.  STORY is sharing your vision, your passion, your understanding, your wisdom, your pain, your joy, yourself.

One simple activity we were asked to do in the etmooc was to create a Six Word Story.  Here is one I created with the app Phoster.

I was also inspired by a video I saw in our Google+ Etmooc Community shared by +Alec Couros.  It was a video of a bicycle journey in Barcelona at the turn of the century.

 It made me think of some video footage that my husband took from a motorized rickshaw in Bangalore, India in 2010.  I spliced the footage together rather quickly and sloppily using Movie Maker and then uploaded it to Youtube - my first ever submission to Youtube - no applause necessary.

My husband named it "Chaotic Passage".  I think it makes an excellent metaphor for this Etmooc learning journey I have been on.  Learning is rarely smooth, there are twists and turns, obstacles and distractions along the way.  And if you're lucky, tu peux voir des vaches dans la rue!