“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.
“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.
“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.
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.
“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.
“When students step out of the door of the institution called school today, they step into a learning environment that is organized in ways radically different than how it once was.”
In spite of the some of the stormy events of the past few weeks; both in the atmosphere and the in political sphere, a lot of really cool things have been going on at our school. Primarily, we have had the chance to engage in some professional learning together in the key areas that are reflected in our School Improvement Plan and we’ve been working in our classrooms to tinker with and implement some of this new learning.
In earlier posts on this site I have referred to the ’3 I’s” of our school plan; Inclusion, Inquiry and Innovation and tried to connect these with our focus on effective mathematics instruction, supporting students with learning challenges and the use of communications technology to support 21st century learning. The video link above is a thought provoking piece on why these ideas are important for our students and our schools.
Over the past few months we have been working in small teams to develop and refine our own questions in these areas; forming teams of 4 to 8 staff members to research and inquire into the ways we can improve our mathematics instruction, understand the different ways that children learn and look at the ways we can use iPads as teaching and learning tools.
Today, a friend and colleague of mine, Dean Shareski, spent some time working with our staff, via a Skype video conference. Dean, who Skyped in from his home in Moose Jaw, Saskatchewan, helped us explore some of the media tools that are available through our partnership with Discovery Education Canada and offered up some practical tips on ways we can use social media tools like Twitter, blogs and Edmodo to support student learning and parent communication.
There was a time when a teacher could believe that they knew everything they needed to know to be successful upon their graduation; those days are no more. It turns out our license to teach is also a license to learn.
“As teachers do we see our role as initiating learners into mathematical communities, speaking and inquiring with young mathematicians at work? Or do we speak to them, trying to transmit a of skills and concepts…developed by previous mathematicians? Are we teaching the history of mathematics rather than mathematics? ~Cathy Fosnot~
It’s encouraging to see the enthusiasm with which our Park Ave. P.S. team has embraced our whole school focus on mathematics teaching and learning~not only because this is an area of personal and professional passion for me; but also because we are, as a staff, uncovering some powerful and important insights about the nature of mathematics and networked learning.
In stressing the importance of inclusion, inquiry and innovation my role has really been that of catalyst and coach; providing resources, structures and guidance for this learning. In our conversations so far, we have discovered that our students have a wider range of skills and, deeper understandings, than the tools that we were previously using; revealed. In our case, all our students, including those students who have been identified with learning disabilities, are revealing capacities and communicating ideas in ways that are both surprising and encouraging.
The problem shown in the photo above (The Sold Out Show) is a great example. In designing the task for her students, Ms M considered the models and strategies her students were using as they solved multi-step multiplication and problems. She created a problem that would push her students to better understand the relationship between these two operations as well as the important mathematical processes of reasoning and proving how they know their solution is accurate.
Rather than demonstrate and have her students memorize the steps toward an accurate solution, Ms M has crafted a problem that will help her students to build and communicate their understanding. The culmination of this task will see the students analyzing and questioning each other’s proofs with Ms M taking the time to highlight the mathematical relationships, ideas and terminology as they are doing so.
For many of us,this is a reversal of the model of instruction we experienced as students; one where the teacher demonstrated the singular procedure and skills that would be needed to complete a task, then assigned a similar task with the expectation that all the students would replicate what had been demonstrated. This model worked, for about half of us.
In our classrooms, we are asking students to show us what they know about mathematics and how they are able to apply this knowledge in context; then using these contexts to push them to a deeper understanding of how mathematics allows us to make sense of our world, communicate about our world and work to solve problems in our world~ a process that the Dutch mathematician Hans Freudenthal describes as ‘mathematizing’.
Creating classrooms where our students mathematize, rather than memorize, is our ultimate goal; as always, your comments and questions are welcome!
“Mathematics is not a careful march down a well-cleared highway, but a journey into a strange wilderness, where the explorers often get lost.” ~W.S. Anglin~
Across Ontario schools are taking a deeper look into the the research and practice on mathematics teaching and learning so our recent work at Park Avenue P.S. in this area doesn’t make us unique. Like all schools in our district, we are part of a learning network; a family of 4 or 5 schools that have agreed to engage in joint work to support and improve our classroom practice. Our network will support our school focus on mathematics teaching and learning from K to grade 8.
As a first step, we have spent some time looking at what our student’s strengths and needs are in this area; using both our own assessments as well as our provincial assessment results from grade 3 and 6 (we will continue this work on our next PA day on Oct. 22nd). In addition, we have also started to read, think and talk about the most current research into effective math teaching; most adults have deeply ingrained experiences that often cloud their perceptions of what math actually is, leading to some confusion at school and home.
Rather than a set of pre-determined rules to be memorized, mathematics is actually a way of structuring and representing our physical world~ like a language. Since the best way to learn any language is to be immersed in it and use it-rather than be forced to memorize it- we are working as a whole staff to design problems and tasks that will help our students do this.
It is also important to note that all learning requires one to struggle and learning math is no different. The struggle we wish for our students is not, however, in the memorization of the mathematical concepts but instead in the development of these concepts. As I used to remind my students when I taught math,; trust what you know and understand, not what you remember.
There are three key areas we will embed into the work we do in our school and network and we are happy to share them:
- Inclusion: all students require support and instruction that draws upon their learning styles, experiences and starting points to construct their understanding of the mathematical strategies, ideas and concepts being taught.
- Innovation: real world applications, models, contexts and tools; designed will form the basis of our learning tasks~and this will look much different that what most of us recall as ‘math instruction’.
- Inquiry: just as most of the math problems we encounter in daily life require us to pose our own questions and often work with others to solve them, our students will be challenged to pose questions, hypothesize and struggle a little to find and prove their answers.
I’m grateful to our colleagues from the Mathematics in the City project, based in New York City, for sharing their research and practices and supporting our learning journey into this new territory and, as always, invite your responses.
“The essence of mathematics is not to make simple things complicated, but to make complicated things simple. ~ S. Gudder~
One of the areas we are going to be applying some focus, effort and time towards this year is mathematics teaching and learning. We know from our own classroom and school assessments that this is an area we need to look more closely at; and this sense is supported by the most recent results from the primary and junior provincial assessment in mathematics.
Most of us have a set of experiences in math that have helped to form our beliefs and feelings towards math, positive or otherwise. It is important to note that our understanding of how to teach math in a way that all students can understand and learn has changed a great deal in the last 20 years. And, as the ‘real’ world has also changed, the need for every student to be able to think critically, solve problems has become an essential part of learning, in math, and every other area of ‘school’.
The video clip at the top does a nice job of communicating our vision of what math class could, and should, look like. And this is the work we will be doing together as a team of professionals, teachers and support staff from JK to grade 8. We also have a responsibility to keep families informed about this learning, and how parents can provide support. This blog is one place parents can go to find posts, pictures and videos that will document, share and connect the learning that is going on at school with our families and school community.
We are all excited about the learning we will be engaging in; the Park Avenue Math Makeover is underway, please feel free to join us as we learn and grow together!
This Friday, the long and eventful career of a great educator and leader will come to an end. I’ve had the pleasure of knowing Diane Muckleston for 15 years, as a colleague, as a friend and, as a member of our district’s math curriculum team under Diane’s skilled and intentional leadership. We were originally introduced by the mathematics consultant of that time, Barry Scully, who thought that we’d make a good addition to the teacher-leaders team that he had helped assemble to support the roll out of the then new Ontario Mathematics Curriculum Document.
We worked together to pilot the use of problem-based math tasks in our classrooms, led after school workshops for teachers and worked on some district writing teams. Soon Diane assumed the consultant’s role that Barry had held and invited me to continue as a member of her math team. Being a new father, I often brought our son Peter, a toddler, to our after school planning meetings and he’d amuse himself with the multitude of colourful mathematics manipulatives. Intermingled with the math talk were conversations about the ups and downs of parenting, with Diane’s boys then approaching adolescence; I benefited greatly from her wisdom and insights.
When Diane assumed the role of District Mathematics Coordinator in our board, I applied to join the team as a consultant and was successful. In the fall of 2003 Diane led a team of 3 consultants (two elementary and one secondary). Over the course of that year we planned and facilitated in-school and after school learning sessions for over 1000 elementary and high school teachers. A year later three additional consultants were added to the team.
With Diane’s leadership our team pushed the boundaries of teacher professional learning; adapting a protocol for teacher joint work through lesson study, integrating technology and blended learning into our work, supporting and facilitating cross panel co-teaching and co-learning for networks of grade 7,8 and 9 teachers in the area of mathematics. The team met around an old rose-pink table that we had rescued from our board’s surplus furniture depot and around that pink table we established a culture of fierce, honest and respectful collaboration that was based upon the belief that every idea was improvable and every perspective was valuable. Everything I learned about the power of trusting, honest practice-based conversations for teachers I learned around that pink table.
The important thing about Diane has always been her skill and focus on developing people and building capacity; from that original team of 6 consultants; 5 are now school administrators and one is head of the math department at her school. As the first cohort began to move on, Diane continued to develop a further wave of leaders who continue to push the boundaries and engage in innovative and creative work with teachers; the consultant who replaced me has just accepted a faculty of education secondment, and so it continues. All this under Diane’s patient, determined and persistent leadership.
So, a couple of weeks ago; to celebrate her retirement, Diane invited all the past and present math team members to her home for dinner. There were over 20 of us, spanning the years of her district leadership, diverse in many ways, but singular in our love of mathematics, our passion for learning and our admiration for our leader. It was a delightful evening of conversation, laughter and catching up. In classic Diane fashion, every detail was considered, the preparation was flawless and the meal; perfection~well almost perfection~ no pink table.
A part of Diane will be embarrassed that I’m posting this to my blog, I know this, but I’m not sorry. I’m grateful for the friendship, the knowledge and the wisdom Diane has shared with me. I’m grateful for the friends and colleagues I have gained as a result of my time working as part of her team. And I’m grateful for the many contributions she has made to our school district.
Too many, perhaps, to count.