Archive for the ‘Mathematics’ Category


February 8, 2014 1 comment

As we engage in re-imagining public education in the coming years, I believe that we must re-think the use of space, the use of time, the structure of the school day and year, the sorting of students by grade, the use of schools within communities and, probably the most significant, the structure and content of curriculum.   Not everything will need to change but it is important to ask the question: “is it right for today or are we doing it this way because we always have?”                                                 ~Ken Thurston~

This past week the Director of our district school board, Ken Thurston, announced that he will be retiring at the end of July. (For my American cousins, in Ontario the title Director is akin to Superintendent). I’ve had the chance to work with Ken in a variety of roles over the 14 years I have known him. He was one of my school superintendents when I was in the classroom, I had the chance to work with him when I was local union steward and committee member and, for the past 4 years I have been proud to serve as a vice principal and principal under Ken’s leadership. It was Ken who sat across the table and led the conversation that resulted in my appointment to the position of principal.

One of key traits I have observed consistently in the years I have known Ken is the importance he places upon relationships. Whether he is thinking about students, staff, parents, unions, community members, trustees, or policy makers, relationships matter most. The other trait I have observed is the willingness Ken has to question the status quo, imagine alternatives and grant agency to those who wish to do likewise.

At Park Avenue PS our students, staff and parents often tell me that I think “differently’ on many issues, sometimes in little ways, sometimes in more radical ways. I do so because we have had a Director not only granting permission, but actually challenging us to do so- Ken’s question, quoted in bold text above, guides my daily work.

During our short time together working as the principal of our school community we have asked ourselves this question and given ourselves permission to re-think our use of tools, time and curriculum. Over the course of our intensive math professional learning this week I challenged each of our grade teams to rethink our concept of how we teach our math curriculum- both in structure and content- and look more closely at how we can use the Landscape of Learning to provide developmental instruction that challenges and meets the needs of all our students.

As we learn how to do this in math, and gain confidence, I’m sure we will find lots of ways to apply this in other areas of our school community and better inform our parents in this area so they can both support and better trust the work we are doing in our classrooms. Re-imagining our school, and re-shaping it to meet the challenges of the world we live in now, is the work that each of us; staff, parents and students, must do together.

Back at our first staff meeting together I remember saying to our staff that I didn’t want to change everything, just the things that weren’t working, and we would make these changes together.  I’m encouraged that, even as he prepares to depart, our Director is giving principals and teachers permission to re-think, re-imagine and re-create our classrooms and schools.

Of course, it’s more than likely I would have continued on this path regardless; but sometimes it is better to have permission rather than have to beg forgiveness.

Balancing the Basics

February 5, 2014 1 comment

Contrary to the stories of doom and gloom that we are hearing, our mathematics education system is not broken. Can we improve what we do? Certainly. Should we throw out the whole thing and go totally back to basics? Absolutely not. There are three key things that can improve what we have – balance, balance, and balance.                        ~Ian VanderBurgh~

I understand the concerns and restless anxieties that parents feel about the performance of our students; I’m the father to 3 high school students who happens to also be an elementary school principal. I also understand the reasons why our media outlets would raise and amplify these concerns- they play a valuable role in shining a light on our democratic, public schools.  It’s a good thing that we are having conversations about these concerns; in our communities, face to face, through the mass media and our social networks.  In that context, I’d like to share some of my concerns…

  • I’m concerned that people are starting to believe that, based upon recent standardized test results, our children are sorely lacking in their mathematical knowledge, when compared with previous generations.
  • I’m concerned that people are starting to see private or commercial tutoring groups  like JUMP Math or Kumon as sustainable solutions to these perceived concerns and beliefs.
  • I’m concerned that people are forming opinions based upon opinions, and not upon what is actually the state of affairs in elementary math education.

This is why I appreciated the perspective that the University of Waterloo’s Ian VanderBurgh shared in a recent op-ed piece in the Globe and Mail.  As a school principal and elementary mathematics specialist teacher I could relate to Ian’s point that community engagement, collaboration between parents, teachers and ‘balance, balance, balance’ are keys to supporting improved student learning in mathematics.

You see, our kids are not the dullards some would have us believe. It would have been great if those of us who occupied classrooms in the 1960’s, 70’s or 80’s had been asked to complete the most recent PISA or EQAO math assessments; it could have provided some baseline data and certainly add some context (and modesty) to the opinions we hold of this generation of children.  I’m certain that the rote learning and memorization that formed the  foundation of my ’70’s era mathematics learning experience would not have prepared me to face the adaptive, open ended tasks that form the core of our current tests. Not sure? Check out some of the questions on the recent PISA assessment.

Nor can we really count on private foundations and organizations to ‘solve’ this crisis. What makes our public schools essential is that they are public; accessible and accountable to all. Not all students and families can access the often costly programs and resources that are touted, which makes it even more important that we ensure that the teachers in our public schools have the knowledge and capacities to teach mathematics effectively to all students. Of course, as one who works in public education, my bias and my beliefs draw me to this stance (just as those who operate private tutoring services are drawn to theirs).

Opinions being what they are, for the most part, instruction in ‘the basics’ is alive and well at our school and this ‘discovery learning’ thing is not the evil, Birkenstock-clad conspiracy that some have opined. We don’t even use the term discovery learning-it’s seems like a rather redundant term- doesn’t all learning require discovery? We recognize that when we involve our students in posing questions and contexts we see greater participation, deeper thinking and more connections to the ‘real’ world. And we recognize that our students need direct instruction on ways to use mathematical models and strategies to help them make sense of numbers and solve problems.

For us, the basics include more than memorization of facts, they include different ways of showing number relationships and arming our students with multiple strategies and tools for solving problems. And we are learning how to better use a balance of direct instruction (teachers teaching) and problem solving (students learning) to do this.


Leaning on PISA

December 8, 2013 3 comments

“Bad news sells” is a very depressing truism of our business, even when the bad news doesn’t remotely convey what’s happening.”        ~Jeffrey Simpson~

The media landscape has been filled with responses to the release of the results from last year’s Programme of International Student Assessment test (PISA). The assessment is designed by the Organization for Economic Co-operation and Development (OECD) to gather data on the core skills of reading, science and mathematics and is administered to a sample of 15 year old students from around the world. The focus of last year’s test was mathematics and, though many countries participated in this assessment, it is important to note some key points:

  • approximately 510 000 students (21,000 from Canada) participated in the assessment world wide and their selection was made by school- all the 15 students at a randomly-selected school would’ve taken the 2 hour test;
  • there was a mix of countries, states and cities that participated; from city-states like Hong Kong and Singapore to countries like the United States and Russia. In fact, the OECD, which administers the test classifies participants as economies and states;
  • the assessment does allow for the gathering of demographic information that permits a more robust and detailed analysis of the results.

The last point is where I will direct my focus for now. Predictably, the release of the test results set off a lively cycle of reaction from the media; with responses ranging from panic, denial to smug self-congratulation. Across Canada media outlets analyzed, sought ‘expert’ insight and opined about the national and provincial results. The initial reaction from the media that I have scanned has been pretty balanced. There were some initial alarms claiming that we are ‘falling’ due to low standards’ a misguided mathematics pedagogy of inquiry and exploration and that we needed to get ‘back to the basics’ and focus on more traditional methods of math instruction. But, as the days have passed and accounting for bias, some helpful points have been raised and discussed:

  • Much was made of the rankings as evidence as indicative of Canada’s declining status in mathematics as we ‘slipped’ from 10th in the world to 13th (out of 65). While, the raw math scores have declined 14 points over the past decade Canadian 15 year olds still perform at high level using this measure. Only muddled math could equate above average as a crisis- especially when one factors in that 4 of the 13 ‘countries’ above Canada in the rankings are actually cities in China that were reported as separate entities.  The key concern here is the decline using this measure and how we can explore this pattern in our context. A common element among the districts that had high performance in math is the emphasis they place upon teacher quality and expertise in the teaching of mathematics. Though factors like curriculum design and socio-economic status play a role; the PISA results confirm that the students who perform best in math have teachers who are well trained, both initially and over the course of their careers.
  • When comparing the performance of Canadian provinces much was made of the superior results in Quebec; with some commentators giving credit to Quebec’s focus on rote memorization and avoidance of the ‘fuzzy math’ that other Canadian provinces have adopted.  Fortunately, commentators have looked more deeply at Quebec and realized that though their curriculum is not that different than the rest of Canada, the investment that they make in preparing and supporting the on-going professional knowledge of their teachers is; with Quebec teachers spending significantly greater time learning about mathematics during their pre-teaching preparation and beyond.  McGill Mathematics Education Professor Annie Savard points out that; “People on their way to becoming math teachers also do plenty of field work, watching and doing hands-on teaching while still in university. By the time they graduate and head into classrooms, they have done a minimum of 700 hours of in-class internships.”   We could also point to Quebec’s decade-old investment in affordable, universal child care as a factor in these results as there is a robust connection between a child’s development of early number concepts and later academic success.

In our school context we are considering the PISA insights to guide both our planning for professional learning and our allocation of resources.  We know that one-off ‘programs’ that emphasize basic skills and memorization do not work just as we know that ‘inquiry learning’ that expects children to discover and develop mathematical understandings by themselves will not work. What we do know is that learning occurs when teachers have the skill to design tasks that require students to struggle, allow them to use models and strategies that they have been taught and compel them to prove and justify their thinking. These are all outcomes of classroom teaching.

As a school, our key investment is in developing the capacities of our teachers to provide focused instruction in mathematics.  And that is why it is exciting that 8 of our teaching staff will spend some time learning about effective mathematics instruction with Dr. Cathy Fosnot over the next week.

Categories: Mathematics

Modelling, not Memorizing Mathematics

November 9, 2013 Leave a comment

“Mathematicians do not study objects, but relations between objects.”       Henri Poincare

I was chatting with a few of our staff this week about the ways we help our students develop their ability to work with numbers, specifically when they are adding and subtracting in the early years.  An important point that I often stress when talking with parents and teachers about mathematics is that the area that many of us believe to be the critical focus of mathematics (the operations of adding, subtracting, multiplying and dividing) is actually of secondary importance.

Simply put- we spend way to much time trying to force children to memorize or learn the operations and not nearly enough time helping them understand the numbers they are using. Put another way- think of the numbers as nouns and the operations as verbs; in math, as in life, there are way more nouns than verbs and they are much more interesting!

This media clip on Using Open Number Lines from Dr. Alex Lawson does a great job explaining how using a model like a number line can support children to think about the quantity value and relationships that exist between numbers in a mathematical situation. It also shows how using models as a precursor to what we call the standard algorithm is important at all stages of mathematical development. For most of us, this type of instruction was just not used when we were learning math in school and it is too bad, because it would have save many of us from a life of math phobia.

A lot of  students and adults think that using the algorithms is the math- it’s part of it, but not nearly the most important part. In fact, the algorithm can most simply be described as a way of showing (or modelling) what has been done with the numbers. A student who uses an algorithm to solve a problem without understanding the relationships between the numbers is no better off than a student that uses a calculator- they both don’t really know what they have done.That’s why we are using models like the open number line- they allow the student to see the connections and relationships between the numbers and build a model of how they can solve a problem.

There are only a few mathematical operations but the numbers are (literally) infinite- the numbers are much more interesting than the operations- tools like the open number line help our students discover and harness this idea.

Evidence of Impact

October 21, 2013 Leave a comment

“Statements without evidence are just opinions – there are too many of those in education and that’s what’s got us into trouble.”       ~John Hattie~


There are a lot of opinions about what our schools should do to fix the problem of of gaps in student achievement. We don’t really have the luxury of entertaining popular theories or opinions- our professionalism requires that we look to the valid research on what represents effective classroom practice. The chart above is an attempt to represent the complexities of effective teaching into a manageable graphic. It is a representation of the massive inquiry that John Hattie, a researcher from New Zealand, conducted to answer the question- when we consider learning for children what influences or factors actually have an impact on academic learning?

If you click on the link it will enlarge and allow you to read it more closely. The inquiry focused on six areas that contribute to learning: the student, the home, the school, the curricula, the teacher, and teaching and learning approaches. A key understanding in reading the graphic is the idea that the longer the slice, the greater the impact on academic learning.  Another understanding in decoding the chart is that the average of all the influences is .40, so any influence  above .40 in the study was deemed to have an above average impact, and any lower a below average impact. There are a few items, like summer vacation and retention of students, that were deemed to have a negative impact on academic learning.

I’ve read the study and took away a few key points:

  • The factors with the greatest impact on student learning are classroom and school-based and involve the interaction between teachers and students
  • Teachers need to reflect upon, on a daily basis, the impact of their actions on student learning and adjust accordingly
  • The giving of feedback to students on their performance only matters if the student understands and acts on the feedback
  • Teachers need to work together to build their understanding of how to design, teach and assess classroom learning; this learning needs to be situated in classrooms and focused on what the students are doing and saying

As you can see from the graphic, some factors have a greater impact than others. As a school community we need to focus our resources and attention on making changes that will have the greatest impact for all our students, but especially those students who need it most urgently.


What is Math?

October 18, 2013 Leave a comment

The problem we really have in math education is not that computers might dumb it down but that we have dumbed down problems right now.         Conrad Wolfram

Staff, parents and students are having many conversations about math at Park Avenue these days…given some of the recent trends in our mathematics performance on the EQAO assessment; these conversations are both timely and important.

We are all entitled to have an opinion about this topic and I’m glad people are sharing their opinions. I came across this TED talk from mathematician/researcher Conrad Wolfram that poses some dramatic and interesting challenges to our traditional views on how mathematics should be taught. I am hoping that those who wish to continue this dialogue will look beyond what they know and believe and open up to the ideas of others; from within our school community and from beyond – including those who are engaged in research like Wolfram.

It turns out that Barbie was right- math is hard. Learning to think mathematically is a difficult as is teaching others to think mathematically.  I am hoping that we can extend the conversations we have started into this forum; to capture some of our ideas and perspectives, the build some understandings on how we can work together as students, families and educators and to push all of our thinking forward.

Our school improvement work isn’t easy, it not always quick, but it is both urgent and important.

Responsibility or Accountability?

September 24, 2013 3 comments

“Do not worry about your difficulties in mathematics, I assure you that mine are greater.”     Albert Einstein


The chart pictured above is a simplified representation of our school’s most recent results from the provincial EQAO assessments our grade 3 and 6 students wrote late last school year. To view the detailed report for Park Avenue PS just click on our 2013 EQAO Report.

The information conveyed from these assessments is used for a myriad of purposes, depending upon the stake and stance of the user. Parents use the school and individual student reports to get a sense of how their child is performing against the established curriculum benchmarks. School and district staff use these results to provide information on student strengths and needs in the areas of reading, writing and math to inform program and professional learning needs. The Ministry of Education uses the results to gather information on the performance of students across the province- to determine program needs and priorities- but also to demonstrate accountability of the system to parents and taxpayers.  And members of the media and other pundits use the results to support and advance opinions and theories on our public education system as a whole.

For us, as a school community of parents, educators and students, these results have given us some things to think about and talk about. It’s not my place, at this time, to spin the numbers. I invite those of you who are interested to read through the report, reflect upon what this information means to you and join us in the conversation about our next steps. The other reason I won’t spin the numbers is because the charts don’t represent numbers, to me they are real kids- the students who spend their days with us at Park Avenue Public School.  At the core, I look to these assessments as one way that our kids can tell us how they are doing; what they are doing well and where they may be struggling.

So, in looking at our assessment results, this year and over time, the trend that we have seen in the area of mathematics is a concern. In spite of what media reports or popular sentiment may be, mathematics is more than just ‘memorizing stuff’ and ‘drilling the basics’. Math is a complex and specific language that is used to describe our both our physical and abstract realities, realities that are based upon patterns and relationships using numbers. However, like any language it can be learned, with direct instruction from adults who know the language and know how to teach it.

There is only one way for our students to receive classroom instruction like this; professional learning for our teachers. No packaged program, no quick fix, no short cuts.  Richard Elmore reminds us that ‘real accountability’ is not found on standardized assessments or tests. Real accountability is the relationship that exists between students, teachers and the classroom tasks that students do each day. Our staff are committed to work together and learn together about how to design and teach tasks that will enable them to provide our students with this instruction in mathematics; from Kindergarten to grade 8.

And it is my responsibility to lead this learning.


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