What's Wrong With How We Are Teaching Math?

Ontario needs to improve teacher training in math, the province’s Education Minister has said in response to standardized test results that revealed students are losing ground in the subject for the fifth year in a row.

CAROLINE ALPHONSO AND ADRIAN MORROW
The Globe and Mail
Published Wednesday, Aug. 28 2013, 11:31 AM EDT

It seems all eyes are focused on Junior Math in our schools this Fall.  Everyone is asking, why are our scores dropping in Math?

For the last several years, the Education Ministry has been encouraging teachers to teach Math through problem-solving using a three-part lesson.  This mandate is grounded in research.  In our board, like many others, our Elementary teachers have been involved in the CIL-M (Collaborative Inquiry and Learning in Math) meaning they have been learning together, through a co-planning, co-teaching, debriefing model to learn how to teach Math through-problem solving using a 3-part lesson.

So why have the scores been dropping?  Over and over again I hear teachers saying "So, when are we going to admit that the 3-part lesson doesn't work?" And I hear parents saying "Teachers need to go back to teaching the basics.  They need to teach Math the way I was taught."

Are they right?  I've been thinking about this question a lot lately, and I have to say "NO!"  The 3-part lesson and teaching through problem-solving WORKS and it is the best method for helping students to develop deep conceptual understanding of mathematical ideas.  So, if I am right, why haven't we seen the scores improve in Junior Math on our EQAO Assessments?

I have several theories that might explain why the scores are not improving.  I believe that our lack of improvement is actually the result of several factors.

1.  Teachers have not fully bought into the 3-part lesson and teaching through problem-solving

• What I have noticed during many of our co-teaching sessions is that teachers have a really difficult time "giving up the floor". The first part of the 3-part lesson, which should be a 15 minute introduction or "Minds On" tends to go on for closer to a half hour of straight teacher talk. This is because old habits are difficult to break and teachers don't fully trust that the learning should and will come out during the third part of the lesson, the Consolidation.  They feel that they have to provide students with enough information at the beginning of the lesson for them to be successful in solving the problem during the "Action" portion of the lesson.  They run out of time in their lesson, and the Consolidation ends up being only 5 minutes long, if it takes place at all.  So while on paper, the plan looks like a 3-part lesson, what is happening in the classroom is much closer to traditional teaching.
• Some teachers see the 3-part lesson and teaching through problem-solving as a "Friday" activity and teach traditional methods during the rest of the week.
• Teachers did not choose to be a part of the CIL-M, they were told to be a part of it.  I've asked many people to explain what their "inquiry" in Math is about and they don't even seem to know that they are inquiring about anything. I've also asked "What is the goal of your PD session today?" I've yet to meet a teacher who is a part of the CIL-M who can answer that question.  They are attending as a form of compliance not because they are hoping to learn something new.  Just like our students can be compliant with a learning task without knowing or understanding the learning goal, so can our teachers!

2. In order for teaching through problem-solving to work effectively, teachers have to be excellent at formative assessment

• In order for teaching through-problem solving to work, the problems have to be in a student's zone of proximal development.  This can only happen if the teacher has good assessment for learning practices and knows exactly what his/her students are ready to learn next.  The teacher has to differentiate for students at different places on the Math learning continuum.  
• Teachers need to be able to determine, through assessment and making learning visible, what students misconceptions are so that they can address them.  This leads to very tailored, explicit instruction, but again, requires that the teacher is an expert at formative assessment practices. 

3. Teachers have to themselves have an excellent understanding of the Math Curriculum 

• Many of our teachers lack confidence in teaching Math.  They don't consider themselves experts in Math, and struggle to identify the "Big Ideas".  Teachers today were taught Math by traditional methods; many of them only have rote learning of mathematical algorithms, and don't have deep understanding of mathematical concepts.  Here is a small example of what I mean: how would you compare 6/7 to 7/8?  When I ask teachers this question, they automatically begin to create equivalent fractions using common denominators.  That's because they don't truly understand fractions.  They should be able to see that each fraction is one piece away from a whole, and the eighths are smaller pieces than the sevenths so 7/8 is the larger fraction.  

So what do we need to do about this?  Well we need to acknowledge that we have a problem of practice.  And saying that we are going to teach Math through problem solving isn't going to truly address that problem.  We need to dig deeper, be more specific.  What do we have to change about our teaching practice to address this learning deficit in Math?

I think we need to do the following:

• We need to assess our students' mathematical understandings BEFORE we begin a unit of study, but we also need to assess them along the way to determine if we are having an impact.  We need to be responsive to our students' needs.  In order to do this effectively, teachers need to develop their formative assessment strategies.  I like Dylan Wiliam's Embedded Formative Assessment and our Ministry's AER Gains site for professional development on formative assessment.
• We need to develop our questioning strategies to provoke thinking, elicit misconceptions, and scaffold student learning to address misconceptions
• We need to use rich multi-step problems that are open-ended or have multiple paths to a solution; that way students have multiple entry points. Dr. Marian Small's resources help teachers to develop rich problems that address the Big Ideas and common misunderstandings that students have.  Here's a PDF that shares Marian's teaching strategies for focusing on the Big Ideas in Math through open tasks. 
• We need to explicitly teach accountable talk in the Math classroom using prompts to promote student-to-student discourse and discussion. Cathy Marks Krpan's Math Expressions is a great resource to help teachers develop student thinking and problem solving through communication. 
• We need to make Math concrete.  Students can't understand the abstract until they understand the concrete.  We have to stop saying,"Here are the manipulatives if you need them." No student wants to say they need something others don't.  We need to say, "Use these manipulatives to prove your solution." Students who are using an algorithm but don't have conceptual understanding will be forced to explore what the concepts really mean. 

Teachers can learn along with their students, but they can also learn with their colleagues through a co-plan, co-teach and debrief model.  Teachers need to get together to look at curriculum expectations along side their student needs before they develop problems.  Once they know what their students need to learn, together they can develop rich problems or activities for their students.  They need to moderate student work together to establish what they now know about what each student knows and can do, and where each student needs to go next in order to achieve the learning goals.   We need to be intentional in everything that we do!

But most importantly, teachers need to recognize that every classroom is a laboratory, and every lesson is an experiment.  We formulate a hypothesis, "if I teach this concept this way, the children will be able to ...." and like any good scientist, we need to observe the reaction; what are students doing?  What are students saying?  Then we judge the evidence.  Were we effective?  What impact did we have on student learning? If we're not effective, we have to create a new hypothesis and determine how to fine tune our teaching to meet the needs of the students in front of us.  

Why Are We Looking at This Data?!?

Percentage of Students at Level 3 and 4
EQAO School Report September 18, 2013

Last year was my first year back in the classroom after working as an Instructional Coach for two years.  I really tried to put into practice all of the strategies I had read and learned about.  It was a wonderful year, the best year of my teaching career thus far.  My students and I had a blast.  In comparing assessments from September to June, I truly believed that my teaching had had a positive impact on my students' learning. 

Then the EQAO scores came out.  I was crushed! I fully recognized that the Junior EQAO Literacy and Numeracy Assessments really are just one type of assessment and can't possibly validly assess all that my students had learned over the past year.  My students had learned to work collaboratively to solve problems, share their thinking, and connect with others outside of their classroom to learn authentically from the world around them.  How can EQAO possibly assess that?  But still, I believed that all of those experiences would have had a positive impact on their ability to be successful on the paper/pencil EQAO test. 

That's probably because one of my very favourite books is "Hooray for Diffendoofer Day" by Dr. Seuss (with the help of Jack Prelutsky and Lane Smith).  In fact, I have always read it to my Grade Six students on the first day of school.  In case you are unfamiliar with it, I've included a YouTube video of the story.  Basically, at Diffendoofer School, the students learn differently, and while they never prepare for the high-stakes test, they are more than ready for it because of their unconventional learning.  Oh - you've just got to watch it - it's still one of my very favourites!

So in my new role as a Curriculum Consultant I was recently at one of the schools I support conducting an Item Analysis of the EQAO data for the Primary and Junior Divisions.  Together with the School Improvement Team, we analyze the EQAO data to determine our students' needs because our student needs tell us where we, as teachers, need to focus our learning.  It is very public knowledge that the scores for Math in the Junior Division across the province are dismally low.  Why?  One of the teachers conducting the Item Analysis with me said "We've been asked to use the 3-Part Lesson in Math and to teach through Problem Solving.  When are we going to accept defeat and acknowledge that it doesn't work?"

And for the first time I had doubts.  I doubted the efficacy of teaching Math through collaborative problem-solving because of the low scores in my own school in Junior Math.  Had I been wrong?  Could everything I had been espousing been incorrect?  I couldn't answer that teacher's question; I couldn't blog - I felt like a fraud.

I needed to take a hard look at the data from the school in which I had taught. What could that data tell me?  What couldn't it tell me?  When we look at the data, we don't only look at the Achievement or Outcome data.  We also have to look at the Contextual or Demographic Data as well as the Perceptual Data.  That is why we shouldn't ever rank schools.  We have a Community Living Classroom in our school and those students are not physically able to participate in EQAO but they are still counted in the overall scores.  We also have a very high ELL population and sometimes have to exempt students who arrive from non-English speaking countries only weeks before testing.  When we looked at the percentages of participating students, our results were a bit better.  When I looked at the scores for my class alone the data looked even better still.  78% of the students in my class achieved benchmark levels in Math. Of course, when you are talking about only 22 students, it is difficult to talk in percentages, but I was relieved none the less. 

I am tired of hearing "these kids can't" and want to prove that these kids most definitely CAN! Last year I became convinced that Carol Dweck was correct in her theory on mindsets.  Basically, she says we can have a growth mindset, where we believe accomplishments are a product of hard work and dedication, or we can have a fixed mindset where we believe intelligence is a fixed entity that can't be changed.  Dweck's research demonstrated that teachers who have a growth mindset are better able to motivate and engage their students.   

Why do I pour over EQAO data?  Because it is one of the best tools we have in this province to reflect on our impact on student learning.  I think my class' EQAO data proves Dweck is correct.  Here's a concrete example.  I had one student who struggled all year in Math.  Early on in the year, she told me she didn't like Math (Perceptual Data).  When I asked her why not, she explained that when she had been in Grade One her teacher had told her she didn't have a brain for Math.  I told her her Grade One teacher was wrong, and I would prove it.  A couple of days ago, I called her at home to celebrate her level 3 on the Math EQAO scores.  She said, "So does that mean I am good at Math?"  I answered, "It means you can be good at anything you want to be!"

So why are the Junior Math scores so low in our province?

I have a theory about the Math.  Traditionally, teachers have always taught the Math lesson to the whole class, then assign a set of questions from the text, and then take up the questions with the whole class.  This teaching strategy has a certain level of effectiveness.  I believe teaching Math through collaborative Problem-Solving is more effective BUT ONLY IF THE TEACHER IS GOOD AT FORMATIVE ASSESSMENT.  If the teacher is teaching through Problem-Solving but does not begin with Assessment FOR Learning, the impact on achievement is lower than the impact of teaching with the text book.  I was able to positively impact my students' learning because I started by finding out what they already knew and what misconceptions they had; then I worked toward closing gaps and correcting misconceptions.  This style of teaching does not work if you don't first teach your students how to communicate their thinking in Math. You have to ask the right questions to elicit their understandings.  You have to know the significance of what they are saying.  If you only ask for an answer, you have no idea how they got there!

During one of the Item Analysis meetings at one of my schools, two of the primary teachers were discussing the following exemplar from the released EQAO Math scoring guides.  

They were having an excellent discussion about whether or not this should be considered a level 3 response.  (By the EQAO scoring system, is was considered a level 30 response).  We ended up discussing what the work told us about the child's understanding of the Math concepts.  Did the child in fact use critical and creative thinking to solve this problem?  What does this child's solution tell us about his/her understanding of Math concepts?  Are we just looking for the right answer?  Or are we assessing the child's level of understanding?

We need to know the kids that we are teaching.  We need to know what they know and what they can do.  We have to give them multiple opportunities to explore Math concepts until they develop deep understanding of these concepts.  

As long as EQAO is out there, I will be pouring over that data, analyzing it, trying to determine what pieces we are missing and how we can do it better.  The current scores in Junior Math tell me that we still need to learn a lot more about how students learn Math.  

I have a fantasy that one day we will live in a world where no one ever says "Oh, I'm not a Math person, I've never been good at Math."

Being Reflective Practitioners

"If we teach today as we taught yesterday, we rob our children of tomorrow..." - John Dewey

I'm currently reading "Instructional Rounds in Education - A Network Approach to Improve Teaching and Learning" by Elizabeth City, Richard Elmore, Sarah Fiarman, and Lee Teitel.

 I actually picked the book up over a year ago because I was intrigued by the title.  As you may know, before going into Education, I was a nurse, and nursing rounds were a huge part of our practice.  So I wanted to know how the authors proposed to use the "rounds" process in the world of Education.

I just wished that I had read the book BEFORE this past school year. Why?  Well, this year, for the first time, I experienced our School Effectiveness District Review process.  This process is mandated by the Ministry of Ontario.  I actually learned so much going through this process, not just about the practice of teaching and learning, but also about human nature, relationships, and the pressure associated with feeling like you are "under the microscope."  I also witnessed the stress teachers experience with the advent of change.

Going through the review process gave the majority of the us the impetus to move our practice forward at a faster pace than we might have otherwise.  That was a good thing.  It also helped us to be much more reflective in what we do, why we do it, and the impact we have on student learning.  I just wish we didn't have to go through a "Review Process" to behave this way!

The first thing that we had to do for the Review Process, was determine a Problem of Practice.  I'd have to say that this was the most difficult part of the whole process. We used our classroom assessment data along with our EQAO data to determine what our Problem of Practice is.  What was interesting was how many people took issue with the term "Problem of Practice".  Many didn't like the insinuation that there WAS a "problem".

This is where Instructional Rounds in Education would have come in handy.  It paints a clear picture of what a "problem of practice" is.  Had I read it before going through the Review Process, perhaps I could have helped alleviate some of the tension.  From my current understanding, a Problem of Practice does not reflect bad teaching.  It simply reflects a need in the school.  For example, we realized that, in general, the students in our school have a very limited vocabulary.  We started there.  But as we continued to reflect, we also realized that the students in our school have difficulty comprehending texts. We wondered, was the reason they had trouble comprehending because their vocabulary was so limited?  Or was it a bigger problem?  Was it that they couldn't make inferences?  Were they having trouble visualizing?  Or is it that overall, they lack background knowledge?  We also noticed that in our Junior Division, our students had trouble solving rich, multi-step math problems.  Should we focus on Math?  Or was the issue with the Math problems actually related to a reading comprehension issue, they couldn't understand the questions?

It was really great to notice the change in the conversations in the hallway.  Teachers were having discussions about teaching metacognition, whether they should do it explicitly in the beginning of the school year, or towards the end and whether or not we were using a common language for Math instruction.

After determining our problem of practice, (we decided to go with reading comprehension), we had to develop an "If... then" statement about something we were going to change in our practice to meet the need we wanted to address.  Again, it would have been helpful to read Instructional Rounds in Education first because it helps outline how to do this.  It defines the "If... then" statement as a "Theory of Action".  Eventually, (it wasn't until JANUARY!) we finally came up with our statement.

The next step was for everyone to participate in professional development so that we could successfully implement the teaching strategies we outlined in our Theory of Action.  The interesting thing I learned from reading Instructional Rounds in Education is that, according to the authors, just because teachers plan together and receive the same PD, it doesn't necessarily follow that what will happen in their respective classrooms will be the same.  After observing classrooms, it became evident to the authors that teachers interpret and implement what they have learned in different ways.

Enter "Intentional Interruption: Breaking Down Learning Barriers to Transform Instruction" by Stephen Katz.  Katz describes the barriers we have as teachers to learning.

I could see all of these barriers in action this year at my school.  It is so hard to get past them.  We tend to look at something new, and say "oh, I'm already doing that, I just call it something else", and not really take the new learning seriously. The other thing I hear most often "that wouldn't work with the students I have this year".

Why is change so threatening?  Why are we so reluctant to teach differently?  These are the questions I hope to tackle this year.

Since we only came up with our Theory of Action in January, and then we had to have PD, we were still novices at implementing the new strategies by June.  We haven't even had a chance to reflect on whether or not they have had an impact on student learning.

As a school, we won't be reviewed again for another five years.  That is just too far away.   It is so important that as educators, we reflect on our practice regularly.  We need to look at our teaching practices critically.  What impact did we have on student learning?  Not just most of the students, but ALL of the students.  Which group of students were we not successful with?  What adjustments do we need to make next?

This seems natural to me.  Maybe that is because of my nursing background.  As nurses, we were each the Primary Care Taker for 10 - 13 patients on the ward (depending on the unit).  As the Primary Care Taker, we were responsible to make decisions about the care for our patients.  Although I had 10 patients, and they all just had some sort of surgery, their needs were very different, and they healed at different rates.  We met weekly to discuss each patient's progress.  If someone wasn't progressing at the rate we thought they should, we worked together as a team to come up with different strategies of care. No one took it personally, no one thought it was a reflection of their nursing practice if someone didn't recover at the predicted rate.  We weren't reluctant to try new strategies - new dressings, new medication pumps, new protocols for healing; we were willing to try anything to be successful, and we documented DAILY the impact of what we did on patient recovery.  We charted the patient's "complaint", our observations, our plan of action, and an evaluation of the impact of our plan.

I think our students deserve that same quality of care, that same reflective practice.  I hope in September, at the first staff meeting, my school will begin by looking at their Theory of Action again and recognize that as a theory, we need to determine what evidence exists, if any, to validate it.  What adjustments need to be made to the theory? How will we dig deeper in the coming school year?

I leave you with the following video that I find so inspirational.  It is time to move from an Industrial Model of Education to a 21st Century Model of Teaching and Learning.  That means we have to be reflective practitioners, practitioners that can define our practice and the impact that we have on student learning.  In the video, Mr. Lichtman says we need to be self-evolving learners and in doing so, we teach our students to also become self-evolving learners.  We can't be afraid to let our practice evolve along with the world as it continues to change at such a rapid rate.  Watch the video - be inspired!

Liebster - Loving All of Those Blogs that Keep the Learning Going!

Liebster: Discover New Blogs

Thank you Sheri Edwards at What Else for including me in your nominations for this fun award. It is a great way to recognize those of us who are “smaller” bloggers —  with fewer than 200 followers. What a great way to help us to connect!  I am thrilled to be included!

Liebster Nomination Rules

1. Link back to the blog that nominated you.

2. Nominate 5-11 blogs with less than 200 followers.

3. Answer the questions posted for you by the nominator.

4. Share 11 random facts about you.

5. Create 11 questions for your nominees.

6. Contact your nominees to inform them of their nomination.

Well, I'm not sure how many followers each of these blogs have.  I am still so new to all of this, I don't know how to find out!  But these are some of my "go-to" blogs that I find myself checking out again and again!

My Nominees:

1. Aviva: Living Avivaloca

2. Rick McCleary:  Mr. McCleary's Musings

3. Heather: Mrs. D's Flight Plan

4. Lori: At the Principal's Office

5. Laurie: Global Grade Threes (classroom blog) Professional Ponderings (Professional Blog)

My answers to Sheri's questions:

1. Why do you blog?

To reflect on what I have learned.  I have always found that in writing, I think.  I don't really write for an audience, but I admit, knowing someone else might be reading what I write pushes me to think harder.  

2. What’s the most important thing a teacher can do for his or her students?
The most important thing a teacher can do for his/her students is build a caring relationship with them. If you build a mutually caring relationship, everything else falls into place.  You will know that child as an individual and do your utmost to help him/her succeed.  The child in turn will trust you and feel safe to take risks. 

3. What’s the most important thing a teacher can do for his or her colleagues?
Same thing, build trusting and caring relationships so that you can help each other grow as professionals.

4. If you could change one physical thing about your classroom, what would it be?
No teacher desk! No student desks!

5. Describe one of your most memorable classroom experiences.
Dancing on the desks with my students after a math test that every single child got an "A" on.  I really believe that if you tailor instruction to a child's needs, everyone can succeed.  

6. What memorable experience do you hope your students have?
No individual memory, I want them to remember how much fun we had all year learning together.

7. How many students/teachers do you have at your school?
There are 285 students and about 14 teachers.  

8. What is your favorite classroom use of technology?
Oh, that's a toughy! I think I'd have to say using the iPads to make movies.  It encompassed so much learning, from researching, writing scripts, filming, editing... they used the iPads for every portion of the video making, and were so engaged.  Every child felt needed and a part of something big.

9. Who/what is your teaching inspiration?
I have many teaching inspirations, but I'd have to say that Twitter has been the biggest inspiration for me this year. 

10. What is 1 teaching goal you have for this school year?
Well, I am moving into a new consulting role.  My number one goal this year is to inspire teachers to open up their classrooms. 

11. In six words, what is your teaching philosophy?
Know your students as individual learners.

Eleven Random Facts About Me
1. I am married to the kindest most patient man in the world
2. I have three adult children who make sure my head never gets too big
3. I have two dogs and a cat who always remind me that it is good that I am here on this earth 
4. I just bought a new Mac and I'm LOVING IT!
5. I love to swim, I think I was a dolphin in a previous life
6. I love to read, read, and read some more, anything, anywhere, any time
7. I believe knowledge is meant to be shared and questioned
8. I love walking on snow that crunches
9. I've learned you can take the girl out of Montreal but you can never take Montreal out of the girl
10. AT 93 years old, I still think my Grandma is cool
11. I'm an extrovert that could easily become a hermit because of the Internet

Eleven Questions for my Nominees'
1. Why did you go into teaching?
2. What do you love most about your job?
3. How do you use technology in your class or school?
4. How many students attend your school?
5. If you could change one thing about Education, what would it be?
6. What do you do after a bad day? 
7. What is one of your proudest moments in Education?
8. Whose blog are you always excited to read and why?
9. What is one of your professional goals for next year? 
10. What is your favourite inspirational quote for Education?
11. Why do you blog?

Making Shift Happen - A Time For Reflection

It is July once again; a time for reflection.

This past year has been the best year I have ever experienced in my teaching career. This is due in large part to the learning that I have experienced this year and the teaching networks I have been a part of. I am more aware than ever that while we are often the only adult in our classroom, teaching is not by any means a solo pursuit, it is meant to be something that is done in community.

The entire paradigm of teaching has shifted. No longer is a teacher the impart-er of knowledge and information; a teacher is now a role model for learning. More than ever this year, I developed a learning stance, and became the chief learner in my classroom, and that made the year fun and exciting for me, and learning a contagion for my students.

Creating strong learning networks that I could go to for advice and ideas was really important to me this year. I am so grateful to all of the very excellent Grade Six teachers in our board that lent me their ears!  I am also grateful to my friends in other boards who acted as objective listeners to my sometimes fantastical ideas. And lastly, I'm extremely grateful to all of the incredible educators I have met in my virtual network via Twitter, Google+ and ETMOOC.

Today I started tutoring a 12 year old boy in Math.  I asked first to look at his report card, and saw that he averaged about a 67% in most subjects. In the Learning Skills section, I noticed that it was stated he needed improvement in "Self-regulation" and "Independent work". I pointed this out to him and asked if he has trouble focusing and paying attention.  He said "Not really". I said "is your class rowdy and do the students fool around a lot?" He said "Pretty much". I asked him to describe a typical Math lesson. He described a class where the teacher taught the lesson at the front of the room for the first half hour, then assigned the questions in the text book for the second half of the period. He mostly ignored the lesson and the questions while he socialized with his friends.

I asked what part of Math he found most difficult, he didn't even hesitate before answering "percentages". I asked him to complete a Proportional Reasoning Diagnostic from our Ministry's Closing the Gap on the Math Gains website. It was apparent he didn't understand the relationship between fractions, decimals, percentages, rates and ratios. We started with factors and multiples, quickly moved to numerators and denominators, then equivalent fractions. I was amazed at how quickly he caught on.  Within one hour he was solving rate problems by converting to a unit rate.  So why didn't he learn all of this in school?

It is no longer okay to teach the way we were taught!  For so many of our students, (I would argue especially our boys), it just DOESN'T WORK!

Although I've had an excellent year in the classroom, and am well aware that I still have so much more to learn, I am leaving the classroom for a temporary hiatus to go back to the Curriculum Department. Hence the name change of my blog! I am going to attempt to Make Shift Happen! We classroom teachers need to change the way we are teaching so that we reach the students most at risk in our classrooms. Every child has a right to succeed and it is our duty to make sure that that happens. I'm hoping that I can support teachers in making that shift happen in their classrooms.

I'm scared though, scared because I know how teachers feel about enforced PD and enforced "Professional Learning Communities". My blog post and the comments that ensued on the PLC vs the PLN (Professional Learning Network) were an eye-opener for me. My challenge this year will be to find a way to be a co-learner with my fellow teachers and support them in making shift happen. I am really hopeful that I can inspire and motivate my colleagues the way I have been inspired and motivated by my PLN. I think the key elements are voice and choice, I just don't know yet how to ensure those elements are embedded in my work with teachers.

I will once again welcome your ideas and opinions. Your comments and emails will be more important than ever to me on this portion of my learning journey.

Life After EQAO - End of Days

It seems as though the year has flown by; most likely because we were moving at break-neck speed to learn the entire Grade Six curriculum before EQAO.

So what does a teacher do with students when EQAO is over and the curriculum has been taught?  We have fun! We have been spending June making movies and having Art lessons.  For Social Studies, the students worked in groups to research a Canadian Explorer, they then wrote scripts to bring their Explorer to "life".  After completing their scripts, they used the iPads to film their movies.  It was so much fun watching them create their costumes and make their props.  Who knew that you could make a ship out of four desks and a blanket?  Or that meter sticks could be your oars?  Next, they worked with iMovie to edit and publish their movies.  Lastly, we watched the movies - this was highly entertaining, especially the "bloopers" they all felt should be included.

Practising Our Filming Techniques

(Here are some pics - I've coloured over their faces for privacy reasons).

A Sailor Spots Land!

William Baffin begging for funding.

In Art, we studied positive and negative space via print-making by watching Youtube videos.  This was one of our favourites:

We simplified this technique somewhat because I didn't want the students working with the Exacto knife.  I purchased inexpensive foam sheets from Michael's Craft store.

Then, we reviewed a simplified "Writing Process":
1. Brainstorm
2. Rough Draft
3. Revise and Edit
4. Publish
I explained that we could actually rename this the "Creative Process" because it is the same whenever we create something, including a piece of artwork.

The students then brainstormed and began creating rough drafts of potential subjects.  This actually took up the bulk of our lesson.  Here are some sample drafts.

 In some cases, the final product changed a great deal from the original drafts.  Some students wanted to draw an image, some wanted to do something more abstract.

Once students were happy with their final draft they copied it onto their piece of foam using a pen to gouge the image into the foam.  Some students opted to cut out sections using scissors similar to the video above.
We used an ink roller and water-based block printing ink/paint that I also found at the craft store and a roller to cover our foam.

Here are some of the results:

This student started off drawing a mustache.  She ended up putting her mustache into the "Big Apple" and decided she should put New York City inside. 

This student wanted to create an image resembling the Artwork we had studied by Aboriginal artists.  He made several prints exploring colour and adjoining the same print over and over again to make different patterns. 

I love having the whole day to do Art.  One or two 40 minute periods is never enough time to really DIG into  artwork - it is just too messy and consuming.  Today, we learned all about charcoal and pencil sketches.  Everyone was engaged, on-task, and learning.  One of my students actually held up a first draft and a final piece and said "look how much I improved from when I started this morning until now".

"Inappropriate Use of Technology"

CC licensed photo  shared by Flickr user Daveynin

Recently, some older students in our school "smuggled" a personal electronic device out into the hallway and they used it inappropriately.  I don't think I need to go into any details, there are hundreds of ways to "use technology inappropriately".

The result is that our principal has decided to ban personal electronic devices from school property.

I understand his frustration and the pressure he must be under.   But I am not convinced this is the solution.

In my class, I have a BYOD policy.  In the past nine months, I had one student bring his phone out to recess because he forgot that it was in his jacket pocket.  He then left his jacket outside, with the phone still in his pocket.  He begged to go back outside to get his jacket.  Luckily, the jacket was still there.  I also had one student take a video of another student who was dancing in a silly manner while we were getting ready for home.  I was in the classroom at the time.  I took his device away for a few days because we have a strict policy in the class that video taping only occurs with teacher permission.

Other than that, I have had absolutely NO issues with their electronic devices.  They come in in the morning, they put their devices in a cupboard which remains locked throughout the day (it also houses our class laptops and iPads).  When we need the devices, we take them out of the cupboard.  It's that simple.

I have done a LOT of EXPLICIT teaching up front about appropriate use of technology and responsible digital citizenship.  My students have a mantra: "What goes on the Internet, stays on the Internet."  They know that they shouldn't post anything they wouldn't be proud to show to their grandmothers.

They also know how to be cautious and protect their personal identity.  They know the dangers.

I don't believe banning electronic devices is the solution.

Let's face it: Kids (and adults for that matter) have been known throughout history to use communication devices inappropriately.  Give a kid a can of spray paint, and he may paint a mural, or he may paint a racial slur.  Give a kid a marker and she might create a title page or she might write her "frenemy's" name on the bathroom wall.  I've picked up a lot of inappropriate notes that have been passed around on paper in my day!  Should we ban paint, markers and paper?  What about the almighty pen?

I know, the scope, and therefore the damage, is so much greater with the Internet available to them.  But as soon as they step off school property they are using the technology.  Are we any less responsible because it happens outside of the school day?  Isn't it better to take our heads out of the sand and teach responsible digital and global citizenship BEFORE the inappropriate usage starts?  Aren't we responsible as their EDUCATORS to teach them how to use digital devices for their benefit rather than to their detriment?

Today I showed my students this TED Talk by Sugata Mitra.  I'm guessing everyone is familiar with it, but in case you aren't, please take the time to watch this video.  More than anything else, this video helped me to understand the power of creating a child-driven learning environment using technology.

I welcome your input on this issue.  It is something we have to face and make decisions about more and more often these days.

More Fun with Grade Six Probability Concepts!

I had so much fun teaching Math today!  We are still working on Probability concepts, and on Friday, I had the students play a game I call "Odds and Evens".  Students work with a partner and they each have a number cube.  They roll their dice and add the sum.  One student chooses to be "odd", the other "even".  If the sum is even, then the "even" person scores a point, and if it is odd, the "odd" person scores a point.

Each pair of students conducted fifty trials.  Then we tallied all of the trials for the class and determined the Experimental Probability.  Next, I showed them how to create a tree diagram for the two dice, and we determined all of the possible outcomes.  Students could see that if you found the sum of the two dice, exactly half of the total number of outcomes were even, (18/36) and half were odd.  So we called this a "fair game" because each student had a 50% chance of winning. 

Today, they played the "odds and evens" game again, but we changed one of the rules.  Once again, they each had a number cube and they played with a partner.  This time, when they rolled the dice, they had to find the product of the two numbers rolled.  Again, one of them had to choose "even" and the other "odd".  I asked them first to predict whether or not they thought this would be a "fair game".  They all believed that, like with the sum game, this would be a fair game.  

It was so fun walking around the room and listening to their conversations as they played.  They weren't long into it when I started hearing:  "This game can't be fair!"  So I walked around the room asking them them questions:

• Do you still think this is a fair game?  Why or why not? How can you tell?
• Why do you think "even" is winning?
• What is your experimental probability? Is it what you expected?

It was so much fun to walk around the room and hear the learning as it was happening.  They were so excited when they discovered that both numbers rolled had to be odd in order to end up with an odd product!

I asked them to conduct 40 trials.  (I like to use different numbers of trials so that they get used to finding percentages with different denominators.)  When they had completed all of their trials, they calculated the Experimental Probability.

I then asked them to find the Theoretical Probability for the experiment.  It was very interesting.  They all made their tree diagrams correctly.  They all knew that there were 36 possible outcomes.  And yet, out of 10 groups, only 2 correctly found the Theoretical Probability.  They all determined that there were 27 even outcomes and only 9  odd outcomes.  But they used 40 as their denominator again.  

The last step in the question asked them to find the Theoretical Probability for 500 trials.  They understood they needed to use equivalent fractions, but as they had been using a denominator of 40, their answers were not quite right.

During the consolidation, I asked one of the groups with the wrong answer to step forward and explain their theoretical probability.  The two groups who did it properly, at first, began to doubt themselves.  Why did most of the class get a different answer than they did?  A lively conversation ensued.  But soon enough, they realized what the majority of the class did wrong - that they had used 40 (the number of trials in our experiment) vs 36 (the total number of possible outcomes) for their denominator. 

I'm hoping that because they constructed their own understanding and clarified their own misconceptions, it will be understood and remembered! 

Teaching Probability Concepts

CC licensed photo  shared by Flickr user Duncan Hall

This week we started our last unit in Math.  It is our Probability unit.  I always feel like this unit is our reward for all of the hard work and amazing learning we've accomplished throughout the year.  This is my favourite unit, and it is usually my students' favourite unit as well.  I'm not a very big text book user on any given day, but for this unit more than any other unit, the text book only makes rare appearances. (I once heard Trevor Brown - course director in Mathematics Education at the OISE/University of Toronto and York University and an Associate Professor at Tyndale University College - say that "Math is not textable".)

I believe that children learn about and develop an understanding of Probability through experience.  For this unit, we play games and conduct experiments. We make observations and we record data.  And then, together, we look at our results and we try to figure out how best to represent what we witnessed mathematically.

Children can't succeed with this unit unless they have a firm grounding in proportional reasoning.  If you put in the time exploring fractions, percentages and decimals; if you make sure that students understand the relationships between fractions, decimals and percentages, then your students will be very successful with Probability.

We are only three days into this unit, and my students keep saying "This unit is so easy".  What they don't realize is that it is only easy because they are using all of their knowledge about proportional reasoning to make sense of Probability.

Yesterday, I taught a lesson adapted from an activity from Marilyn Burns' Math Solutions course.  I love this activity because it teaches students how to compare the Theoretical Probability to the Experimental Probability. Here is the original experiment.   I adapted it by having my students compare the results of two different spinners.  Each of the two spinners had three colours, but on the first spinner, red represented half of the spinner, and the other two colours each represented a quarter.  On the second spinner the three colours each represented a third of the spinner.

For our "Minds On", we determined the Theoretical Probability for each spinner as a fraction.  (Theoretical Probabilty = Favourable Outcome/All Possible Outcomes).  Then we turned that fraction into a percentage.  Once it was a percentage, it was easy to determine how what the Theoretical Probability would be for 100 trials for each colour on each spinner.  Using equivalent fractions, we discussed how the numerator would change depending on the number of trials.  Then we discussed how the percentage probability could be written as a decimal and placed on a number line where 0 is "impossible" and 1 is "certain".

I had the students write their predictions for the different number of trials on their personal white boards so I could quickly gauge who was understanding and who was not.

Then came the fun part - the Action.  We had ten pairs of students working together.  (One thing I've learned over the years: make your group sizes according to how many jobs there are to do.  One recorder and one spinner means two kids in a group).  Each group did fifty trials on each spinner.  They recorded the results on grid paper I had made especially for the activity. It looked like this:

You can see the students' individual whiteboards lying on their desks.  We use these all the time. They use them instead of "scratch" paper.  But mostly we use them for quick formative assessment. 

Once everyone had completed fifty trials for each spinner, the students cut out their strips and we attached all of the reds together, (they cut out strips but leave the "x's" on.  We glue subsequent strips via the "x"), then all of the blues, then all of the yellows keeping the results from the two different spinners separate.  (The children said they looked like long strips of ticker tape.)  In total, we had done 500 trials for each spinner.  Looking at the long strips we taped to the wall, it was apparent that for the first spinner, the red strip was twice as long as the blue and yellow strips, but for the second spinner, all three strips were just about the same length.

It took a long time for the students to complete all of the trials.  But taking the time was worth it.  This morning we started our math lesson with the Consolidation of yesterday's experiment.  It wasn't long before they were able to explain to me that the more trials they did, the closer the Experimental Probability approached the Theoretical Probability, and as one of my students shouted, "that is just so cool!"

We were ready for a challenge after that!  So for our Action today, I gave the students percentages for certain colours, and they had to work in groups to devise spinners that would result in the correct Theoretical Probability represented by those percentages.  In a sense, it was working backwards from what they had done yesterday.  They quickly realized that making spinners with 10 sections was the easiest way to complete the task since they had been given percentages.

As an Exit Ticket, I gave them an Open Response questions from last year's EQAO Assessment.  My students found this question "easy".

Exit Ticket: Connor has a bag of coloured tiles. There are 1 green, 3 black, 5 blue and 6 red tiles. He reaches into the bag and chooses 1 tile without looking. What is the probability that the tile is not red?
Justify your answer.


Show the value of the probability on the number line below.

0 ____________________________________________________________________ 1

I am loving teaching this unit.  When I take the time to create lessons that I enjoy and can be enthusiastic about, my students also enjoy the lesson and share my enthusiasm.  When we are all having fun, everyone learns!

EQAO Musings

The countdown has begun; there are only seven teaching days left until we begin our EQAO assessment.

I know there are many teachers out there who are very against EQAO testing, and I fully understand their arguments.  I recently read a post by Andrew Campbell that gave me plenty to consider on the whole EQAO question.

However, notwithstanding Mr. Campbell's points, I believe that the EQAO provides educators with some very useful and powerful information.  Taken as a whole, it does show us patterns of achievement so that we can see -  as a teacher, as a school, and as a board - where we need to change our practice.  It is appalling that our minority groups don't perform as well as their majority counterparts.  But that just tells me there is more work to be done in closing the achievement gaps for these students.

When comparing two schools with similar demographics, it is interesting to note when one school is outperforming the other, (which happens often).  As a teacher, I want to know: What are they doing in that school?  Is it something I can and should be doing with my students?

I'm ashamed to admit this, but knowing that the EQAO test is looming over us, I am motivated to never give up on my students.  Case in point - we recently had a Measurement test in Math.  A full third of my students struggled with finding the area and volume of triangular prisms.  If I didn't know that they would be tested on these concepts in June, I might have said "oh well, they'll learn it again next year".  But that EQAO test makes me accountable.  It is not acceptable for 33% of my students to not understand these concepts.  So I reassessed my teaching, and retaught the unit to these students using different teaching strategies.  The EQAO assessment provides me the impetus to keep on trying. I shouldn't need that impetus, but I do.

It is fairly common practice for teachers to teach students "frameworks" for answering Open Response questions on the EQAO.  But I've had experience scoring (they are "scored" not "marked") the EQAO assessments, and I discovered that these "frameworks" don't work.  Typically, the students who used a four-square model to answer Math questions didn't do well.  The curriculum suggests that we teach students to use a 4-part problem solving model.  This model works, forcing students to use a framework does not.  Just because they draw a four-square on their paper doesn't mean that they will be able to work through the four steps of problem solving.  They actually have to have deep understanding of the Math concepts to answer the Open Response questions.

Likewise, I've seen many different frameworks used when answering the Reading Open Response questions, such as "point, proof, comment" or "APE" (Answer, Proof, Extension).  These don't work either.  In order for the students to be able to respond to the questions appropriately, they have to comprehend the text.  Nothing else works.

So when I hear people complain that teachers are just teaching to the test, it bothers me.  In my experience, taking time to teach students frameworks for their Open Responses is a waste of time.  We need to spend every moment helping our students to develop deep understanding of the content areas. We need our students to be able to construct their own understandings of Math concepts and written texts.  That is the trick to helping them succeed.  

I have found over the years, that while the EQAO assessment might not be perfect, and it might not be scored perfectly, it does do one thing very well.  It assesses how well we are teaching the Ontario curriculum. It has forced me to teach that curriculum better every year and to constantly improve my practice so that everyone in my class succeeds.  I just wish it could happen during the very last week of school, because trying to teach 10  months of curriculum in 9 months does not seem fair or just to me or my students.  And I've learned the hard way that you have to teach ALL of the Science and Social Studies curriculum as well because often the Reading and Writing activities are based on the content areas.  Without those under their belts, the students don't have the necessary background knowledge to succeed.  These days, we are moving at break-neck speed; and that isn't any fun at all.

Blended Learning in Their Eyes

I finally got around to having a class discussion about Blended Learning.  I wanted to hear from my students: what did they think of Blended Learning? Do they like it?  Do they find it helpful?

This is what they had to say:

• Most of kids don't always pay attention to the teacher; but kids pay attention to what is on the Internet, so if you didn't pay attention in class, you can learn what you missed on the D2L (our learning management system). ~ one of my English Language Learners
• I like Blended Learning because it can help you get ready for tests, everything you need to know for the test is on the D2L.
• Using the D2L is helpful because if you forgot a text book at school and you have homework, you can just message your friends and ask them to email you the work.
• I like the D2L because I work better on the computer; I have trouble spelling and using punctuation so doing my work on the computer and submitting it through the Dropbox means I can do my work better.
• If you are sick and you miss school, you can just go on the D2L and find out what you missed.
• Kids mostly use the Internet and video games for entertainment these days, so when you learn from the D2L it is more like entertainment and doesn't feel like school. This particular comment was reiterated over and over again, especially by the boys in my class.
• Kids pay more attention to what is in front of them, using digital tools makes kids pay more attention.

Overwhelmingly, the students in my class love Blended Learning.  They are already expressing their concerns that next year, with a new teacher, Blended Learning might not be an option.  They find learning this way more engaging, and it places them in the driver's seat, giving them more control of when and how they learn.  Using a learning management system, like the D2L, they now have greater recourse when they don't understand something. 

We've had a lot of fun experimenting with Blended Learning this year.  My students asked me, back in January, to "flip" my class.  A flipped class is when you provide the introduction to a lesson via a video or link on-line that students can access at home as homework prior to a lesson.  It gives them background knowledge, so that when they come to class, they are ready to start the "action" or "working on it" part of the lesson right away, while the teacher is there to provide support.  So often, teachers stand in front of the class lecturing, and then students have to do the practice work alone at home where they don't have the teacher's support if they run into difficulty.  The idea with a flipped class is that the students get the "lecture" at home, and then they work on the practice at school while the teacher is present and able to scaffold for them. 

Some of my students had heard about flipped classrooms and they wanted to try it out.  It didn't work that well for us.  Invariably, some students didn't do their homework, i.e. didn't watch the video or access the links, and other students didn't understand the videos or links.  So I still had to address the information one way or another prior to the "action" part of our lesson.  But it wasn't a total failure.  The students who didn't understand the videos went home and watched them again AFTER the lesson, and said that they made sense now.  They realized that they need some "background" knowledge to understand the videos.  Likewise, I think the lesson went more quickly and made more sense to the students who had watched the videos prior to the lessons.

What we learned was that it is helpful to have the information available for the students to access on-line both prior to and following lessons.  That is what I have continued to do all year.  Some students choose to always access the information prior to the lesson so that they are "primed" for learning.  Others prefer to access the information after the lesson to "top up" their learning.  Either way, it puts the students in charge of their learning and that is the goal.  They are paying attention to whether or not they understand.  They are monitoring their learning. 

I believe strongly that every teacher should somehow incorporate Blended Learning into their teaching.  If you teach in Ontario, your board can probably get you access to the Desire 2 Learn (D2L) learning management system that I have been using with my students.  But if you don't, Edmodo is another great option.  It looks similar to Facebook, and allows you to input your students into classes and groups so that they can access content and submit work electronically.  

But even if you are not ready for a learning management system, you can start small.  You can create a class blog where you share information that is happening in class, upload videos for your students to access, and links for them to use to supplement their learning.  There are so many on-line supports for learning, it seems wasteful not to give our students easy access to them. And if you are intimidated by the thought of managing a blog, please don't be.  I knew NOTHING about creating and managing a blog last June.  I taught myself using on-line supports via the "help" button and Youtube. Trust me, no one can be less computer savvy than I was last June. 

Consider having your students use some digital tools to collaborate together, like Prezi, Google Docs, or iMovie.  Many students are already familiar with these tools and are using them for fun at home.  Encourage a student to become the "expert" on a tool and teach the rest of the class how to use it.  You don't have to be comfortable with these tools for your students to use them successfully.  

The greatest thing about Blended Learning to me is that it is not only more engaging for the students, it makes teaching much more fun as well!