Tuesday, October 31, 2017

Complex instruction that helps ALL students

When I was a 5th and 6th Grade homeroom teacher, I always taught maths in homeroom groups, though I'm aware that in many schools maths is a subject that is often streamed or set, with high-ability students being taught in separate classes.  I was interested to read what Jo Boaler writes about teaching heterogeneous groups, as her research seems to back up the observations I made from my own experience.   She describes a pedagogical approach called "complex instruction", devised by Liz Cohen and Rachel Lotan, to make group work equal.  This has implications for other subjects as well - so it can be used with any grade level or subject.

Complex instruction is basically made of up 4 interrelated ideas:  multidimensionality, roles, student responsibility and assigning competence.  Let's think about each of these and how they contribute to higher student performance in heterogeneous groupings:

Jo first starts of describing what a one-dimensional math class looks like - one where there is only one way to be successful and where the focus is on procedures.  These are classrooms where some students rise to the top and others sink to the bottom.  And yet, Boaler points out, mathematics is a broad and multidimensional subject.  This can be encouraged by assigning open-ended and challenging real-world tasks that are difficult to solve alone, to mixed groups of students - the idea behind this is that there are many different approaches that can be used and that these differences are shared with the class so that all students benefit from seeing how problems can be solved in different ways.  Following a multidimensional approach shows students that there are many ways to be successful with maths.

Along with assigning rich tasks to groups, complex instruction involves giving the students in the groups various roles.  This shows that each person has a part to play, and every few weeks the students change groups and are assigned a different role.

Assigning Competence
This was an interesting concept for me.  It involves raising the status of students who think they may be of lower status in the groups by praising something they have said or done and bringing it to the group's or whole class's attention.

Teaching students to be responsible for each other's learning
This involves intentionally teaching students how to work in groups (listening to each other, respecting different viewpoints etc) and letting the group know that one member of the group will be asked about the work of the whole group and will be asked about the mathematical concepts the group was working on.  As the other students in the group would not be able to help the student being asked, it was the responsibility of everyone in the group to make sure that all group members understood the concept.  Jo Boaler writes that with this approach students start to see mathematics as a collaborative, shared pursuit that is all about helping each other and working together.

How does this help the high-achievers?  Boaler writes that many students identified as high achievers are simply procedurally fast, but often they have not learned to think deeply about their ideas, explain their work or to see mathematics from different perspectives. Boaler noticed that while these students initially complained about working in groups and having to explain their work, they soon started to appreciate this approach because it gave them to chance to explain their thinking, which helped to consolidate their own understanding.  In fact their learning accelerated when compared to students who were tracked, because explaining their work took their understanding to new levels.  She writes, "Many of them had come in as fast, procedural workers, and the push to work with more breadth and depth helped their achievement enormously."  Even more importantly, she writes "Neither the high nor the low achievers would be as helped if they were grouped only with similar achieving students."

Sunday, October 29, 2017

True mathematics engagement

When I became an elementary homeroom teacher I knew that I was going to be responsible for teaching maths - which given my own experience at school was quite a daunting prospect.  As mentioned before, I was brought up on SMP in the UK and this didn't really give me the background for teaching a more conventional approach to the subject.  Looking back now, however, I think my own schooling really benefitted me because it made me think hard about how I learned, what had worked for me and what hadn't, and I was able to deviate away from the Addison Wesley textbook to incorporate the ideas that I had that would bring maths alive for my students.

One example was probability.  We started this unit playing lots of games of chance - using dice, spinners, coins and so on.  I wanted the students to really get into the topic before we started to delve into the maths.  During these classes we talked about what makes a fair game, and we also talked about the concept of theoretical and experimental probability.  In pairs, students then had to design a game that could be played by students throughout the primary school.  They had to make sure that the game was fair, and also that they knew what the chances were of winning.  We set up our "Chance Encounters" game fair in the foyer of the school and classes booked to come and play the game.  The students recorded how many played the game and how many times the students won the game, and then back in class again they looked at their games and tried to work out why the experimental probability did or did not match with what the maths would have predicted.  You can see some more examples of the games students designed here.

Another time, along with reading the story of Gulliver's Travels, we looked at scale.  Students made artefacts to scale based on the various lands where Gulliver found himself, for example in Lilliput where the people were only 6 inches tall, and in Brobdingnag where the people were giants.  What would a stamp look like in Brobdingnag, for example, or a fork in Lilliput?  During this same unit we also made some scale models of the school and we went on to use geometry to look at designing buildings around the world (click here to see student examples).  We called this unit Designing Spaces and it involved visualizing, planning and building. Students used geometry to analyse buildings from around the world, to design and build their own house models, and to create plans for their designs. Rather than studying mathematics, the students became mathematicians, engaging in a form of mathematical thinking that is applied in all societies to design living spaces to meet people's needs and to make sense of the physical environment.

In both the above examples the students were engaged in various tasks that would develop a mathematical mindset - they learned about the true nature of mathematics in a practical real-world way.  And they were excited!  Jo Boaler writes, "Interestingly I found that mathematics excitement looks exactly the same for struggling 11 year olds, as it does for high flying students in top universities - it combines curiosity, connection making, challenge, and creativity and usually involves collaboration.  These for me are the 5Cs of mathematics engagement."

It has been fun today for me to look back at these student projects from 1998 (almost 20 years ago - wow!) and to reflect on how all those years ago, before we'd even heard of Jo Boaler, my students were engaged in inquiry in maths and were certainly excited to use the 5Cs of mathematical thinking in their learning.

Saturday, October 28, 2017

Pursuing dreams and enhancing the lives of others

I've always loved our school's mission statement, which among other things states:
We inspire all of our students ... to pursue their dreams and enhance the lives of others.
I was thinking about those words this morning when I read the following sentence in The Innovator's Mindset:  "Dreaming is important, but until we create the conditions where innovation in education flourishes, those dreams will not become a reality."  I also reflected on one of the "what if" questions:
What if everyone in our organization, not just our students, was encouraged to pursue his or her dreams?
I asked myself, is my school one where everyone is encouraged to pursue his or her dreams and enhance the lives of others?

As I'm at my mother's in the UK right now I've been watching a fair bit of television and I was really interested in the BBC programme The Ganges with Sue Perkins.  In Episode 1 Sue took a trip to the source of the Ganges, and in Episode 2 she was at the holy city of Varanasi.  Watching these programmes reminded me of a colleague at ASB who decided last year to become the first person ever to kayak the whole length of the Ganges - going from the Gangotri Glacier at the source of the Ganges to the Bay of Bengal at its mouth.  I know that when Brendon asked the school for the time off to do this journey he spoke about the ASB's mission and about how this was a dream that he wanted to pursue.  On his Ganges 2016 Facebook page he wrote:
Why am I doing it?
A challenge really. First and foremost because I am a Physical Education teacher and I consistently teach students about goal setting. One day I wondered whether I was practicing what I was preaching. I was always encouraging students to aim high, believe in themselves, extend themselves. To put themselves out there, reach for their true potential. The question was, was I? Hence this challenge.
Another part of our school's mission was also alluded to by Brendon as he wrote, "In India at the moment there is a major focus on cleaning up the Ganges river to enhance the lives of those living around and supported by the river. I thought it would be fantastic to travel along the river and see what changes were happening and hopefully bring attention to the changes being made."

George Couros writes: "In a place where every learner is encouraged to reach his or her dreams, these "what ifs" can become reality."  ASB is certainly such a place!

Anyone interested in knowing more about Brendon's journey can visit the Facebook page Ganges Source to Sea.

Photo is dawn on the Ganges, taken by me at Varanasi in October 2015

Creating a culture of innovation

More thoughts from The Innovator's Mindset:

George Couros writes that to create a culture of innovation you first have to focus on learning and growth, and he has a list of 8 things to look for in today's classroom that will help schools to achieve a culture of innovation:

  1. Voice - learning is social and co-constructing knowledge empowers learners
  2. Choice - students need input into how they learn and what they learn
  3. Reflection - it's important for learners to take the time to think about and understand what they are learning
  4. Opportunities for innovation - innovation shouldn't be a one-off or special event, but should become the norm
  5. Critical thinking - encouraging students to ask questions both about the information they are finding and also so that they are empowered to challenge the ideas of others to help everyone move forward
  6. Problem solvers/finders - students who find problems gain a sense of purpose in solving something authentic
  7. Self assessment - this can provide another opportunity for reflection, as students can assess themselves.  George writes, "I think we spend too much time documenting what students know and not enough time empowering them to invest in their own learning and helping them understand their strengths and areas of growth."
  8. Connecting - with experts and with an authentic audience.

Friday, October 27, 2017

No place for thinking in math class?

I can't remember how I was taught maths when I was in primary school, but I vividly remember the maths that I was taught from the age of 11 onwards.  It was called SMP maths, and was developed by a group of researchers in the 1960s who worked out of Southampton University.  I recently found out that SMP was a response to the call for a reform in mathematics teaching, following the launch of Sputnik by the USSR.  It's kind of incredible to me now that the maths I was learning in the 1970s was a response to the Cold War and Space Race!

When I was 11 I had no idea what SMP was.  At school we called it "Stupid Maths Problems" and I do remember having to do problem after problem after problem.  SMP was abandoned in the UK in the 1980s when the National Curriculum was introduced, however before this schools were free to set their own curricula and buy the resources they wanted to support that.  Therefore at school I didn't learn about algebra, trigonometry or geometry, instead I learned about things like graph theory, non-cartesian co-ordinate systems, vectors and non-decimal number systems (I remember binary - I can't remember much about any of the others).  The aim of SMP was to improve the mathematics curriculum taught in the UK, but now it has been criticised as putting a whole generation off mathematics by trying to dive into abstraction too early.  SMP did not really fit in with the exam system either.  I remember sitting my maths O'level at the age of 16 and having to take my shoelace out of my shoe and use it to make a sort of scale model of the problem I was trying to solve that involved the circumference of a circle. However, reading Jo Boaler's book Mathematical Mindsets, I realise that in many ways I was fortunate as I was not drilled in maths facts.  Jo, who was at school in the UK at a similar time to myself, writes that her school was focused on the whole child, and as such she also didn't have to memorise tables of addition, subtraction and multiplication facts.  Instead she learned number sense, which is the deep understanding of numbers and the ways they relate to each other.

As mentioned in a previous post, Jo argues that maths facts are held in the working memory section of the brain, and that under pressure (for example during timed tests) the working memory becomes blocked, causing anxiety.  This is what puts students off maths, and Jo claims, is leading the the maths crisis that is currently being faced in both the UK and the USA.  Instead of memorising facts, Jo writes that we should be offering conceptual activities that help students understand numbers.  There is brain research that shows that the left side of the brain handles facts and technical information and that the right side handles visual and spatial information - and the learning of maths is optimised when both parts of the brain work together to develop new brain pathways.  Jo writes,
The more we emphasize memorization to students, the less willing they become to think about numbers and their relations and to use and develop number sense.
Jo goes on to compare the learning of maths with the learning of English.  In order to understand novels or poetry students need to know the meaning of many words, yet we are not teaching the fast memorization and recall of hundreds of words when we teach English - instead we take words and use them in many different situations - talking, reading and writing.

Getting back to the SMP, and indeed to the Addison Wesley mathematics text books that I was presented with when I became a homeroom teacher in Grades 5 and 6, the thing I remember the most were pages and pages of practice questions and worksheets that had to be completed every day.   I was interested to read that Jo states that we do not need to practice methods over and over again - what we really need is to reinforce ideas by using them in new ways.  Jo writes about repetitive practice:
We do not need students to take a single method and practice it over and over again.  That is not mathematics; it does not give students the knowledge of ideas, concepts, and relationships that make up expert mathematics performance ... The oversimplification of mathematics and the practice of methods through isolated simplified procedures is part of the reason we have widespread failure in the United States and the United Kingdom.  It is part of the reason that students do not develop mathematical mindsets; they do not see their role as thinking and sense making; rather they see it as taking methods and repeating them.  Students are led to think there is no place for thinking in math class.
I was also interested to read the PISA results that show the lowest scoring maths students in the world are those who use memorisation, whereas the highest scoring students are those who think about the big ideas and the connections between them.  Clearly what students need is to be given interesting situations and encouragement to make sense of them.  In this way, Jo points out, "they will see mathematics different, as not a closed, fixed body of knowledge but an open landscape that they can explore, asking questions and thinking about relationships."

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The importance of compression when learning maths

I've been reading Jo Boaler's book Mathematical Mindsets and have become really interested in the idea of compression.  This is not a brain process I've heard much about before, but it's described in the following way:
When you learn a new area of mathematics that you know nothing about, it takes up a large space in your brain, as you need to think hard about how it works and how the ideas relate to other ideas. But the mathematics you have learned before and know well, such as addition, takes up a small, compact space in our brain.  You can use it easily without thinking about it.  The process of compression happens because the brain is a highly complex organ with many things to control, and it can focus on only a few uncompressed ideas at one time.  Ideas that are known well are compressed and filed away.
What this means is that when necessary you can recall the maths quickly, to use as a step in another mental process.  However many students see maths as a bit of a slog because they are not engaging in compression - and the reason for this is because maths is often taught as rules and methods, and not as concepts.  The brain can only compress concepts and not rules - hence students who learn the rules have to struggle to hold onto them - they are unable to be compressed, organized and filed away for later use.

This is why it's important to help students approach mathematics conceptually at all times - and the conceptual understanding of maths is what Jo Boaler refers to as a mathematical mindset. This also explains why in Making the PYP Happen it states: "In the PYP, the mathematics component of the curriculum should be driven by concepts and skills rather than by content."

Jo Boaler is a British mathematics educator and is currently Professor of Mathematics Education at the Stanford Graduate School of Education.  Her website is Youcubed.

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Some more thoughts on innovation and leadership

My Grade 6 class in 1996
I'm now in Part II of The Innovator's Mindset and am thinking about innovation and leadership based on the following quote:
As leaders in education, our job is not to control those whom we serve but to unleash their talent.
As I read this it reminded me of  the reasons why I moved to my school 6 years ago.  Before moving to India I was not in a school that was much interested in innovation, and so the thing that really attracted me to ASB was its "can do" attitude.  I'd never been in a school before that had an R&D department, and I was amazed by the culture of yes.  To this end, George Couros writes:
The problem is that when you say "no" to innovation for any reason - people feel reluctant to attempt trying new things in the future.  Their thinking is, "If I am not allowed to do something that could impact learning in my classroom or other classrooms, what purpose do I have in serving the needs of the school as a whole?"  In other words they think, "My ideas don't matter."
My 5th Grade students and their families
at our first student-led conference in 1999
When teachers feel like this they really have just 2 options - not to bother to try anything new, or to go ahead and try it and then ask for forgiveness.  Many, many years ago, when I was a 5th Grade teacher, my teaching partner and I decided it would be really useful to have students at the parent-teacher conferences.  We felt that it would be good for students to be part of the discussions (after all it was their learning) and we felt that they could best demonstrate and articulate their own progress and goals they would like to set for the future.  We knew we were on shaky ground - nobody had ever had students at the parent-teacher conferences before - but we ploughed ahead anyway.  Now having 3-way or student led conferences is commonplace, but it was sad that we had to do it in the way we did, as we feared we would not get the permission to go ahead if we asked for it upfront.  George Couros writes about this in the following way:
If we've established a culture in which educators feel their only option is to ask forgiveness for trying new things, this is not an educator issue, it's a leadership issue ... squashing the ambitions of those who want to go above and beyond to try something new will ensure schools have only "pockets of innovation" at best, and, at worse, no innovation.
Looking back now, I'm aware that my colleague and I did some really wild things.  For example we went and dug up the paving stones in the school playground and then gave students the task of trying to use simple machines to bring the stones up to the 2nd floor of the school through the windows in order to try to demonstrate the technology that the ancient Egyptians used in building the pyramids.  Another time we dug up the school playground and each class buried a set of artefacts, which the other class then dug up, taking on the roles of archaeologists and trying to work out what culture we represented.  I remember this was the first time we had students video and document the whole process and then we posted this on a class website - and this was at a time when our school didn't even have an internet connection back in 1996!  And yet I'm aware that we were just one of those "pockets of innovation" that George referred to.  Personally I think we did really great things in Grade 6 - and over 20 years later I'm still friends on Facebook with some of those 6th Grade students - as well as with my teaching colleague - but I'm also aware that Grade 6 was seen as being a bit "out there" and that although we had a class website we didn't really have a way of interacting with other educators around the world.  I know that change can happen one person at a time - but it could have happened much quicker and had more impact if we had been more connected.  I have to say though that I very much appreciated that school for tolerating the "crazy" ideas of two 6th Grade teachers!

(By the way I just checked to see if that old 6th Grade website is still there - it is!  Amazing!  The photo above is this 6th Grade class in 1996 - my first class who ever published their work on the internet.  Were we innovators?  Yes, I think we were!)

Thursday, October 26, 2017

George Couros's 8 characteristics of the innovator's mindset

Here are some of my highlights based on reading George Couros's book The Innovator's Mindset, along with some of my own thoughts.  As more tech positions in schools are morphing into innovation and coaching positions, I'm interested to see how much of what George identifies is applicable to myself and my own future in education.

  • Empathetic:  empathetic teachers think about the classroom environment and learning opportunities from the point of view of the student, not the teacher.  George's question is, "Would you want to be a learner in your own classroom?"
  • Problem finders/solvers: traditional schools pose questions/problems to students but in real life there is often no step by step way of finding the answer.  George writes "Sometimes it takes several attempts and iterations to solve real-life problems, and, sometimes, there are several correct answers ... Ewan McIntosh notes that finding the problem is an essential part of learning - one that students miss out on when we pose the problems to them first ... Sometimes teachers need to lead from the front.  Other times, our students' learning experiences are improved when we move alongside them or simply get out of the way."
  • Risk takers: as teachers we know that not everything we try will work with every learner, yet George writes, "Risk is necessary to ensure that we are meeting the needs of each unique student.  Some respond well to one way of learning, while others need a different method or format.  Not taking the risk to find the best approach for each student might seem less daunting than trying new things, but maintaining the status quo may have dire consequences for our students."
  • Networked: for years bloggers such as George have argued that being in spaces where people actively share ideas makes us smarter - and social media provides the place for ideas to spread.
  • Observant: this ties in with the last point as George writes that "sometimes the most valuable thing you get from the network isn't an idea, but the inspiration or courage to try something new."
  • Creators: anyone can consume information but that doesn't equate to learning - learning is creation not consumption ... knowledge is something a learner creates.
  • Resilient: for me this is a really important one - certainly in my time in tech I've noticed that anything new and different can see threatening.  George writes, "for those with an innovator's mindset, the reality is that their work will constantly be questions simply because it is something new ... innovators must be prepared to move forward, even when the risk of rejection is involved."
  • Reflective: we need to be asking ourselves what worked and what didn't, what we would change and what questions we have as we continue to move forward.
So here is the crux of the matter:  as leaders we cannot tell others they should be innovative while we continue to do the same thing.  As I reflect on the 8 characteristics outlined by George, there are some I know I'm strong at (being networked, risk-taking, resilient and reflective) and others that I could do with more work on (creating, being observant, empathetic - actually I think I am empathetic to students, what I need is to become more empathetic to the challenges my colleagues are facing - and being a problem-finder).  

I'm at the end of Part I of the book.  It's been really useful to me to reflect on some of the questions that have been posed and to think about where I am today.  Now it's time for Part II which focuses on creating the conditions that empower a culture of learning and innovation.

If you want to purchase The Innovator's Mindset you can find it on Amazon.

Creating our own education: voice, choice and ownership

I have been reading on in George's Couros's book The Innovator's Mindset, and at the start of Chapter 2 I have come across a quote from Stephen Downes:
We need to move beyond the idea that an education is something that is provided for us and towards the idea that an education is something that we create for ourselves.
The reason this struck me as being so important is because I've recently finished Cognitive Coaching Days 5-8 (again) where I was able to again think about the concept of efficacy and the belief in the ability to succeed or to make a difference, and more recently I've been considering the changes that will come to the PYP with greater emphasis on learner agency.  As Downes alludes to, people with agency take responsibility and ownership of their own education/learning.  Dispositions that we want to promote in both students and adults who have agency include critical and creative thinking, perseverance, independence and confidence.  This idea of perseverence is built upon by George Couros in Chapter 2 of his book when he writes:
Having the freedom to fail is important to innovation.  But even more important to the process are the traits of resilience and grit.  Resiliency is the ability to come back after a defeat or unsuccessful attempt.  Grit is resolve or strength of character.  These two characteristics need to be continuously developed as we look for new and better ways to serve our students.
I think the word "serve" is important here.  I've come across this again recently in my reading around coaching - a coach is there to serve the coachee, to help the coachee tap into his or her inner resources, to help convey a person to where he or she wants to go.   In the same way that a coach needs to give up his or her own agenda, teachers need to be mindful of the diverse needs and interests of the students they serve.  George suggests some questions that will help teachers focus on the students (as opposed to the curriculum, standards etc), for example
  • What is best for this student?
  • What is this student's passion?
He is a real advocate of developing empathy for our students and of pursuing our desire as educators of giving our students the best that we can.  An innovator's mindset, therefore, is focused on the desire to create something better - and to do that we need to continually ask the question "is there a better way of doing this?"

Back to the quote at the start of this post, if education is something that we create then both teachers and students are partners in the process.  As teachers we need to work alongside the students, to monitor their learning and to provide feedback and feedforward to help the learning process.  We need to listen to their wonderings, and help them to explore their interests through open-ended tasks and we need to involve students in making decisions about what and how they learn - which means as teachers we often need to co-learn with our students.  In these inquiry-based classrooms, our teaching should be "just in time" and not "just in case".

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Saturday, October 21, 2017

Teaching kids real maths

I've been doing some research about teaching mathematics, and decided to start by reading and listening to the thoughts of the British technologist Conrad Wolfram, who for a number of years has been arguing that we need to rebuild a maths curriculum for the computer age and that students should be calculating "just like everyone does in the real world".  His argument is that school maths is very disconnected from the maths used to solve problems in the real world, and that it needs to be more practical, more conceptual and less mechanical.

I watched his TEDtalk where he states that maths is more important to the world than at any point in human history, yet at the same time there is falling interest in mathematics education, and a lot of this is because we are not teaching "real" maths in schools.  He argues that maths isn't something that is just done by mathematicians, it's done by geologists, engineers, biologists and so on, often using modelling and simulation, yet in education it is mostly being taught using "dumbed-down problems that involve lots of calculating, mostly by hand".

Conrad Wolfram talks about how mathematicss education should basically be done in 4 steps.
  • Posing the right question
  • Taking a real world problem and turning it into a maths problem
  • Computation
  • Taking the answer back to the real world and seeing if it answers the question
The real issue with the way maths is taught in schools today, argues Wolfram, is that most of the time is spent on step 3 - probably about 80% of the class time - and we are teaching students to do step 3 by hand despite the fact that this is the step that computers can do much better than any human.  He argues that we should be teaching students how to do steps 1, 2 and 4 which involves conceptualising problems and applying them, and teaching students how to use computers to do the computation.  He says
Math has been liberated from calculating. But that math liberation didn't get into education yet.
As Wolfram sees it, the problem is not that computers dumb down maths education, but that without them we can only pose dumbed-down problems to students right now.  We don't need to have students work through lots of examples in order to come to an understanding of mathematical concepts, what we really need to do is to teach students to understand how maths works, and the best way to do that is to teach programming, which makes maths both more conceptual and more practical.

Wolfram also argues that using computers in maths allows us to reorder the curriculum.  We currently teach according to how difficult something is to calculate - but he says we need to change this so that we reorder according to how difficult it is to understand the concepts - the calculating can be done by computer.  He talks about moving from the knowledge economy to the "computational knowledge economy" and this can only be done by a "completely renewed, changed maths curriculum built from the ground up" and based on using computers to perform the calculations.  Hearing this, as someone who really believes that technology can transform learning, I became very excited indeed!

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