Mathematics is as much an aspect of culture as it is a collection of algorithms. ~Carl Boyer~
One of the things I admire about the teaching staff I work with and lead is their willingness to take risks and adapt. I think it’s really important that kids spend their time with adults who care about them and have a high expectations; and these two concepts are not mutually exclusive. We are well under way on our journey of school-wide transformation in math teaching and learning and are at the point where those ‘pockets of practice’ that were evident in some classes are now evident in all our classes. Parents are seeing their children using models and strategies that seem strange and unusual to them and we are getting questions, lots of questions.
Most of the questions or concerns we hear are based upon the lack of understanding of how mathematics teaching has changed over the past 20 years and how these changes have been received by parents and the general population. Part of my job as principal is to help people understand our practice and our pedagogy so let me try to address a few of these concerns:
- ‘The New Math’ There is no ‘new math’. Math is the language we use to understand and describe the patterns, relationships and characteristics of our universe.This language is expressed using numbers and symbols that have remained constant for thousands of years and will remain so as long as the fundamental physics of our universe remain the same. We can use a lot of terms to describe math, but new is not one them folks. The emphasis in mathematics has always been on understanding number patterns and relationships to think and reason, this is far from a new phenomenon.
- So What is New? Over the past 30 years a few things have changed where it concerns education; in math and all other disciplines. We now expect that schools will ensure that all students meet a high standard of literacy and mathematical understanding (see Employability Skills Index), In addition, research into the neurological, psychological, and sociological factors around learning have had a profound impact on the pedagogy and teaching practices of teachers. In other words, we know we are capable of, even though we may not all be capable of it yet.
- The ‘Real Basics’ Often, parents struggle to understand the diversity of models and strategies that our teachers are introducing and question why we aren’t teaching the basics. By basics they usually mean things like the standard procedures for addition, subtraction, multiplication and division- also called algorithms. Anyone who has tried to actually explain the algorithm for long division without using tricks or vampire analogies (just what is a goezinta anyway?) knows that an algorithm is anything but ‘basic’. The real basics are the numbers, and our emphasis on helping students understand our number system using models and strategies that make sense to them allow them to use mathematics in its truest form; a powerful, logical language for solving problems and communicating rather than a set of clever tricks and short cuts. If a child doesn’t understand the numbers they are working with, they don’t know the math. It is also important to note that since they are culturally based, there are actually many algorithms, more than those of us who experienced a western education can even fathom.
Across our school, we are working together as a team of educators to better understand and teach our curriculum in a way that will enable all our students to become mathematically capable. Not an easy task but ultimately a worthy one. At its core, mathematics is a language that is expressed using numbers- the beauty of which is the infinite nature of these numbers, not unlike the infinite capacity of our students.
“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 ~
Our supply of those rolls of brown craft paper are depleting rapidly here at Park Avenue PS and with good reason. Over the past year we have looked deeply into the structures and design of our classroom learning environments to ensure they reflect the best practices of universal design for learning and have a degree coherence and consistency across the grades. As a result, one might notice now that our K-8 classroom environments share some common characteristics such as desks or tables arranged in groups to support students working in teams and more open floor space, learning tools and materials stored in a more accessible manner and an intentional use of the walls as visual supports for learning.
Traditionally, classroom walls have been used to display completed student work, more often than not student art work, or written pieces completed by all the students, While the intention to acknowledge and celebrate tasks that have been completed is noble, the question that begs to be asked is how does displaying learning that has already happened help a student who is struggling with what is being learned now? Rather than being a static archive of what has been learned, the walls of the effective classroom need to be an evolving, active documentation of what is being learned.
Education researchers refer to the use of charts and images showing the learning goals, components of a successful task and anchor charts showing the meaning of the strategies and terminology; as essential components of an effective classroom- or ‘high yield’ teaching approaches. And this, is where the rolls of craft paper have become so helpful.
One of our highly experienced Special Education Teachers, Anita Simpson, is creating, along with her students, a Math Wall (it literally fills a wall) that represents the key Big Ideas, Models and Strategies from the Mathematics Landscape of Learning she and her students are exploring. As you can see in the photo above, the wall shows a record of the strategies that students have learned and will need to use along with the models and ideas that connect with these strategies. In Anita’s class, students can be seen glancing at the wall to check the meaning of terms, remind them of strategies or to explore the relationships between the ideas, models and strategies. The wall serves as an anchor chart and road map that is visible for all.
With exception of the ideas, strategies and models labels, the wall was blank in September. Together with her students, Anita has carefully documented the learning on the wall-it’s an impressive sight. So impressive that similar walls are popping up in classrooms all over the school.
Over my 2 plus years as principal at Park Avenue I have stressed the importance for us to develop a set of coherent, common practices in mathematics teaching to support student learning. Mathematics is at it’s core a language; and tools like Anita’s math wall allow our students to immerse themselves in this language while they are engaged in meaningful problem solving- which is the core of a comprehensive math program.
More craft paper, anyone?
“Every accomplishment starts with the decision to try.” anonymous
My colleague and mentor (from afar) Cathy Fosnot describes traditional math instruction as’ “teaching math as if it were a dead language.” rather than the living, dynamic and fluid field of study that it actually is when it is taught effectively. As a teacher, consultant and now, principal, I’ve spent most of the past 20 plus years trying to help students understand mathematics this way while trying to defend this practice to parents and other skeptics (including, quite often, my own colleagues).
It was with appreciation and a sense of relief that I viewed the short video (above) that our York Region District Mathematics Curriculum Team created last spring. The video was posted to YouTube with the intention of defining what effective math instruction should look like, sound like and feel like for all of the students in our district, from kindergarten to grade 12, and help communicate the components of an effective, comprehensive mathematics program to our community and stakeholders.
Unlike the math instruction many of us recall (insert unpleasant memories here) a comprehensive math program (CMP) is a synthesis of meaningful problems (drawn from real contexts), teacher-led mini-lessons (based upon the struggles students are encountering) and games and puzzles (to support student curiosity and make connections to real life). These three components form the basis of the math instruction we provide at Park Avenue PS and I am really proud of the manner in which all of our teachers have embraced this framework.
The component where we are applying the most focus at this time is the mini-lesson- a 5-8 minute lesson designed to build student understanding of our number system along with the mathematical models and strategies that students can use when they are solving meaningful problems. In the photos below one can see two examples mini-lessons
The photo on the left shows one of our grade 4 teachers showing the whole class some of the different models, or tools, students can use to solve and prove the answer to a 3 digit from 3 digit subtraction problem and stressing with the students the importance of using a model and strategy that they understand. In the photo on the right, Our grade 1/2 teacher is guiding a small group students to use diagrams to keep track of the quantities of numbers they are using in a 1 digit from 1 digit subtraction problem. Notable is the use of an erasable whiteboard (sorry, no work sheets here) and the use of talk; teacher to student as well as student to student, as the anchors of the mini-lesson.
In most cases, the biggest problems are not solved with grand, sweeping efforts but through the steady application of effort over time. We are seeing the impact of mini-lessons as it is changing the way our students think, reason and prove in mathematics and, more importantly, the way they feel about mathematics. By breaking the complexities of mathematics into accessible mini-lessons we are giving our students the both tools and the confidence to try- I applaud the work of our teachers and students for making this decision and appreciate our district math team from providing the structure of the comprehensive math program to guide this work.
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.
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.
“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.