How we weave the web of numeracy concepts

In 2018, I started to write a blog post about my thoughts around teaching Mathematics in stand-alone units. I didn’t ever publish the post, but I found it in my drafts – when I was thinking, again (still) about the same thing this week.

The idea that any concept in numeracy “stands alone” is one that really bothers me. Jo Boaler,  our numeracy heroine, says, “Mathematics is NOT a group of unconnected topics, it’s like a web of interrelated areas that can be used to explain the world around us.” She has created a series of these ‘webs’ that help us to visualise the connections between concepts. They help us to see how impossible it is to think about mathematical constructs in isolation. The interplay between the concepts is what allows us to apply what we know to what we are learning.

Image is taken from You Cubed 

The idea then, that we might teach a unit of maths without connecting it to other concepts is impossible. How would it be that, for example,  place value –  understanding how the position of a digit indicates its value and a number’s relationship to other numbers in the base ten system – is not connected to a host of other concepts? 

Some schools use a spiral approach to teach concepts – touching on them briefly each week. Is this better than stand-alone units? How would we get children to dig their teeth into learning if they had so many things to juggle at once, for such a short time? 

What about if we embrace these ideas?

1. Nothing in numeracy really stands alone. 
2. Modelling how we notice and name the concepts that we touch on in a lesson/unit, so that children do the same will remind them (and us) to draw on what we know to work out what we don’t know. 
3. Immersing our learners  in ‘investigations’ regularly – learning engagements that are applicable to real-life contexts that draw on numerous mathematical concepts- will make the connections visible and authentic.
4. Units may need to be taught as ‘stand alone’ to allow time and space for deep inquiry, but our choice of number talks, low threshold high ceiling provocations and learning engagements that have an ‘open middle’ will assist us in making connections explicit for our learners and for us to be gathering continuous, meaningful data to help up plan.

I would love to know how other people help their mathematicians become more aware of the interrelated nature of concepts in mathematics.

Who will we be when we grow up?

Deciding what to choose to study or which career path to follow after school might be as simple as a ‘feeling’ or as daunting as a muddled, murky puddle. 

My 18-year-old, Year 12 daughter is somewhere in between. She knows that she is interested in and passionate about  Education, but she does not have that directing ‘feeling’ that she definitely should do the teaching.

So, as the ‘responsible adult’, I suggest that she speaks to people who have worked in the different professions that she is considering. ‘Of course!’ she agrees, “I want to talk to a teacher!” Not wanting to point out the obvious, I think about what kind of ‘teacher’ she might want to talk to and I connect her with a young, vibrant, talented and intuitive lady who has her own class for the first time this year. 

After their conversion, my daughter is exhilarated and is bubbling over with thoughts that they have shared. “What’s your big take away?” I ask and I am blown away!

• To make a decision, think about what are your values and what is important to you. There is no right or wrong answer.
• Think about what kind of person you want to be. Who do you want to be?
• The world is going to be different when you come out on the other end – so don’t chase what job you want- go based on what’s important to you- that will never change.

We have had many conversations since ‘the inspirational conversation’ and I have been led to think about what other advice I could add to these thoughts. 

My list includes : 

• Be true to yourself. Jobs change, colleagues, the world and your friends change too. The one constant is that you are always with yourself. 
• Think big. Don’t consider your choice as a life long commitment. Jobs, work places and careers are multifaceted. Study something that will feed your brain and heart while  nurturing your curiosity. 
• Studying is a life long journey. This is just the beginning. Think about what you enjoy, what inspires you. 
• Think about role models in your life. What qualities would you like to develop that you have seen in yourself and in others? 
• Have dreams and aspirations. Reach for the stars! 
• Find your spark. As Sir Ken Robinson said in one of his Ted Talks, ‘Sometimes what you are looking for is not on the surface, it’s buried deep’. Dig deep, find what you love and make sure to include this in what you do everyday. 

What advice would you give someone choosing a study path?

What advice do you wish you had received before you chose what to study?

Agency and online learning

Student agency is by no means a new concept. It is one that has been toyed with for generations. As Sam Sheratt said on his blog in 2018 “Progressive and innovative educators have been doing some of these ideas for years. Schools have been designed around them. Movements have evolved around them. Books have been written about them.”

So what have we noticed about agency while we have been learning online, during COVID-19?

Have we noticed ….

• Students intuitively ask questions that get them caught up in seeking knowledge and furthering their skills, no matter where they are learning?
• Planning and teaching responsively rather than planning a series of lessons which we work through sequentially, provides richer, more meaningful and successful lessons, despite online platforms?
• Provocation is a powerful tool that immerses learners into the depth of curiosity and often the throes of productive struggle (a concept that Jo Boaler often shares)?
• Seeking to broaden understanding rather than gather reams of facts creates the platform for life long learners?
• Immersing learners in engagements that require critical thinking skills (that children perceive as being ‘fun’ and ‘playful’) allows them to develop essential skills that are transferable to all learning disciplines and all areas of their lives?
• Transdisciplinary learning is the most meaningful vehicle to empower students to fill their ‘mental toolbox’ so that they can draw on any skill, in any other learning area, rather than confining them to a ‘subject’?
• Teaching responsively is the same as ‘individualizing learning’ and serves as an effective strategy, no matter what the context is?
• When we name and notice actions, skills, dispositions and progress, students do the same and they strive to own and emulate this practise?

Have we noticed that when we ensure we make what we notice integral in teaching and learning, learner agency is developed in an authentic way?

Should we be working one on one with children?

What is the value of working one on one with a child?

M started coming to me, once a week, for 45 minutes, 3 years ago. At the time, his mindset was that he could not and would not do maths. I listened carefully to what he told me. This included:

• “I am bad at maths.”
  “Maths is too hard.”
• “If I try and get it wrong, I am stupid.”
  “Everyone else knows how, but I can’t get it.”

In the beginning, the trade-off for engaging with numbers was that we had to connect them to something that he could make. We experimented with numerous slime recipes. He would go and find new recipes and I would lose the half measuring cup. He would mix colours and I would refer to the ratio of blue to red that he was mixing. But slowly, M forgot to ask about what we could make and he stopped looking for distractions.

 

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If I were to give M a standardized test of any sort, he would probably still not have much success. However, I now notice he is able to

• Engage with numbers for an extended period of time
• Think about estimates that are realistic
• Manipulate “friendly” numbers mentally
• Make connections to what he knows
• Draw on tools and materials that he knows are useful (hundreds charts, a calculator and counters are top of his list)
• Display an awareness of when he has a fixed mindset and even laugh at himself
• Celebrate when he has (even small) successes

For children who find maths a real challenge, strategies for helping them to develop number sense include:

• Constant revision of concepts, in varied, meaningful contexts
• Noticing and naming the strategies that they are using
• Modeling numerous ways of doing things – until they find the one that has meaning for them
• Finding friendly, competitive games in which maths concepts are practised, without this being the focus
• Assessing when they can be pushed and when they have had enough
• Never underestimating them
• Making reflection an integral part of every session
• A multi-sensory approach
• Showing them how to make the calculator and other technology their friend
• Acknowledge their fear and anxiety and celebrate every success
• Find their passions and interests and show them how maths is a part of these

Although we only get to work together once a week, M and I have a shared goal… to help him become a well rounded citizen. One who can operate independently – make comparisons, estimates and take care of his own needs in the world. I guess this is the goal for all children – no matter how competent they are at understanding maths concepts. However, knowing how M learns best, I am constantly reminded that one on one mediation is required; in addition to paired, group and whole class learning.

The important thing about learning at exhibition time …

The important thing about learning is that it is erratic, unpredictable and considerably different for individuals. The thing about the last few days of the school year is that everything usually starts to feel like it is winding down. This is not really true at a PYP school, where learning, creating and taking action is the focus; particularly in Year 6, where the exhibition is so close, it can almost be touched. Even at this point, there is still time to change our thinking, collaborate and question. This is true for student learners, who are documenting, sharing and remodelling;  and teacher learners – who are talking, blogging, reading …

The focus of the exhibition is the journey, and in observing and engaging with children, it is clear that action – in its many forms – has been the driving force behind our exhibition unit.

Children have sketchnoted their learning journey and will use this as their prompt for sharing with their audience on the night of their exhibition. Each learner has a different story to tell. Each one has demonstrated this throughout the course of their adventure. Learning and action are considerably different for each of them.

They each create a headline – a simple slogan or phrase that will catch people’s attention, summarising the gist and impact of what they have learnt.

The “formal event” of the exhibition – the part before individuals will get to talk about their journey- will be shared in the form of “creative expression groups”. These groups comprise a variety of children, from different classes, who will tell the story of the journey through song, dance, rhythm and images. It is not a “polished performance’ – it is a child centred, created and shared celebration of one element of the exhibition process. These are  the key areas about which the groups will share:

1. Beyond Ourselves
2. Conference
3. Inspiration
4. Collaboration
5. Primary Sources & Mentors
6. Dispositions
7. Learning Journey
8. Creativity
9. Action
10. Jewish Studies Connections

The important thing about learning is that it is erratic, unpredictable and considerably different for individuals. The thing about our exhibition journey is that it is all of this. This is what student agency, differentiation, conferencing, explicit teaching at points of need and modelling look like. The important thing about learning is that it is real!

 

A successful provocation

The ‘Beyond Ourselves’ 2018 Conference is over. The invitation to the Sharing the Planet unit is shared. Our hope is that it has sparked an interest and that children would have had the opportunity to broaden their understanding of the Sustainable Development Goals and of how each person’s contribution makes a difference in the world.

The day, broken up into 3 workshops, 3 sets of inspirational speeches and a reflection group has sparked much interest – from students, teachers and external presenters.

Workshop leaders from external organisations have expressed their gratitude for being included in our day.

 

Children who wanted to share 5-minute inspirational speeches did so. Today, they are buzzing with pride. M, who shared her thoughts about treating all people, especially those with physical disabilities, as equals cannot stop saying “Thank you” for the opportunities that the day presented.

The masters and mistress of ceremonies for the speeches are basking in their success.

P, a second language English speaker who, although excited, was really nervous to speak, said, “I feel like I have climbed Everest. I was so scared, but I did it!”

Workshop participants are eager to thank workshop leaders and talk about what they have been exposed to.

Reflection groups towards the end of the day were powerful. The planning that the group leaders did, served them well. Many chose to use reflection cards as a check-in, some posed questions – all felt prepared and excited to take on the role of leaders. Teachers were there to be part of the group and hear different perspectives about the day.

One leader, however, expressed some disappointment in his sessions. “The reflection group wasn’t the way it was meant to be. The teacher took over and we couldn’t do what we planned.” To me, this is an achievement! Learner agency is no longer a “thing”. It is a living concept that children understand and desire.
Our staff was invested and willing to do whatever it took to make the day a success. The children were open-minded and engaged. We have all been exposed to a whole range of organisations, people and ideas that go ‘Beyond Ourselves’. Learners are now free to explore and develop their understanding and awareness of whatever they wish, to see that every small thought can lead to action that impacts way beyond themselves.

 

Thank you, Dear Teacher …

Dear Teacher,

Thank you for taking the time to get to know me. For considering who I am. Where I come from and why I seek attention.

Thank you for giving me the extra time I need to think before you call on someone else to answer your questions. Thank you for stopping people from waving their hands in the air so that I don’t think that they know the answers and I don’t have to think.

Thank you for remembering that although you know what you know, I may not thrive and learn in the way in which you teach. Thank you for asking me questions that help me inquire and think rather than telling what to change and how to change it.

Thank you for remembering that I have a family. That sometimes I feel sad and may not contribute in the way you would hope. Thank you for not giving up on me at these times – for starting each day with a fresh slate.

Thank you for setting the scene so that I am able to recognise my successes and work on the things that challenge me.

Thank you for giving me opportunities to make mistakes without you pointing them out and expecting me to be like everyone else.

Thank you for arriving at school when you are sad and letting me know that it’s okay to be “not okay” – and for showing me how people can still get through the day, even when things are rough.

Thank you for recognising that I may not be ready to learn what you are ready to teach. Thank you for acknowledging that you only learnt some things as an adult and that there is still a whole lot you do not know.

Thank you for remembering that I may not feel comfortable to sit in the position that you think is best, that I may need to eat when I am hungry and share a thought with a friend at a time that you may not think is opportune.

Thank you, dear teacher, for your dedication, kindness, care, guidance and support. Through your modelling of unconditional care,  valuing learning, and being the best person you can be, I am learning to accept myself as a learner and a very special individual!

Documenting small group learning

Selecting children to be part of a withdrawal maths group was a process. As a rule, I am not in favour of withdrawing children to “support” or “enhance” their learning. However, sometimes, rules need to be broken. The group of 5 children was very carefully selected.

As a team, we thought about:

• How might the children benefit?
• How might this create a stigma?
• What might children “miss” when they are out of the class?
• What might they gain when they are out of class?
• Is the gender imbalance in the group an issue?
• What is the focus?
• How will we measure progress?

I spent a great deal of time unpacking, planning and convincing people.

Initially, two of my identified learners were unsure whether they wanted to buy in. I asked for 1 week (2 lessons) before they made up their minds and they agreed.

In the first session, we established the “why” and the “how”. The “what” was to be a work in a progress – a co-constructed development of content and skills that we would develop.

Today’s session begins with a quick counting game as we walk to our room.

“Start at 197 and add 3 each time.”

No one questions why we were are counting and walking. No one puzzles when the number is changed, a decade is crossed or a decimal is thrown in.

There is a  pre-algebraic puzzle on the table as they enter and each child grabs a pencil and starts to tackle the thinking in their own way.

We share strategies (there are two different solutions) and as we go along, the children say things like, “Oh, I know where I went wrong!” and “Wait, a minute, I want to change something.”

I ask the children to think of their learning on a continuum and place themselves on a number line (0 is “I still understand the same as the last time we did something like this” and 10 is “I have made 100% progress).

Each child plots their growth on their own number line and we decide to make a human number line. One child says,”We need a ruler to measure the length of the room!” and he grabs one, measures 3 meters, tells us where the middle is and we organise ourselves.

 

The excitement is the room is infectious! We all celebrate when each child takes their number (tweaked with a decimal point) and creates two questions for another person.

“What is 0.6 more than my number?”
“What is 1.4 less than my number?”

There is a moment of tension when the question, “What is 0 .9 more than my number [7.6] ?” is asked. We all sit and think about our strategy and then we use a tweaked version of Jo Boaler’s  Number Talk protocol to share our strategies.

 

Each session is carefully documented, focussed on the children’s confusion, misconceptions, progress and learning. Today’s lesson ends with an exit card:
“What did you learn/think about OR what confused you / do you want to know more about?” Their answers will guide me in planning next week’s session.

I am blown away by the children who started coming to this group , a few weeks ago, because they need “extra assistance” (and the two who have forgotten they needed convincing) – their mindset and willingness to engage and learn is ensuring that they are developing their self-belief, ability to reason and  articulate their thinking and their desire to learn something that they see the relevance of.

 

Shifting from teacher to learner …

Extracting “big ideas” is the key – how can we apply what we learn to other things?

I spent a week out of my teaching role, in the role of the student learner for a week. Much of what I was engaged in was challenging and I felt uncomfortable. I found myself going for walks, getting water and chatting with my peers. These experiences always serve as reminders of how our learners feel every day.
The transferable ideas that I am reminded of from this learning experience are applicable to all ages, in all learning areas and in any language. There are things that I want to remember about my learners and there are things that I want to remember for my teaching. Most are things that are applicable to learners and teachers, sometimes simultaneously, sometimes not.

At times we need to think with like-minded people. At times we need to think with people who think in completely different ways.	Theory can be interesting. When absolutely necessary, telling leaners what we (teachers)  know needs to be in small bursts and broken up by them talking and thinking to construct meaning.
Giving learners choice is vital –  in little things (where to sit; when to go for a walk; how to take notes) and big things (what they will do with what they are learning; how they will construct meaning).	“Just a minute, let me think” is a crucial mantra. Saying what we want to say, just as they start to think is detrimental.
Metacognition is important … making connections and seeing the relevance of what we are learning have great importance too.	Learners who feel insecure about what they know need structure. Giving too much structure to others is stifling. Find the balance.
We can grow and develop our brains – we need to believe this and challenge ourselves in order to do so.	Providing the opportunity for all students to be challenged is a fundamental part of what we do.
Human interaction (otherwise known as mediation) is the most important thing in learning. Rich language, feedback, guidance and reflection form the basis of all learning.	The instrument/ tool/content that we use to guide learners is secondary. The skills which they gain, that can be applied to different areas is primary.

Reflection has been vital in my understanding and application of what I learnt.

Do I provide enough time and space for meaningful reflection so that learning becomes relevant to my learners?

New maths assessment required

Formal assessment for the sake of collecting data makes no sense to me.

I am opposed to anything that is the cause of unnecessary stress and anxiety to learners; generates extra work for teachers and does nothing to inform learning and teaching.

I am, however, in favour of collecting data to see “the big picture” when it comes to learners. Everything is an assessment and ongoing, purposeful, formative assessment, when linked to meaningful, personal reflection, is key to planning for learning that is directed and owned by learners, and is the director of explicit instruction and guidance from teachers.

We use the Early Numeracy Interview in the younger years for all children, for Number Strands.

From Year 4 to Year 6, we use it for certain children, who are identified as having some “gaps in their knowledge” to help us identify where to direct explicit support. We are noticing that many children do not have  “formal methods” for solving equations by the time they get to Year 6. Secondary school teachers expect all children to solve maths questions using THE formal method.

While we spend a great deal of time using apparatus and encouraging children to draw and explore a variety of methods, we still have some children who do not have a recognised “formal” method to draw on.  Do they need this? Or do they need a successful, efficient and effective way to solve questions using all 4 operations?

Having identified a handful of Year 6 children who need more assistance than what they are currently receiving in maths, I decide it was time to draw up an interview that would allow me to see exactly what their understanding of place value is and how they interpret real-world problems (can they identify the correct operation to use) and which strategy (if any) they use to solve these problems. Knowing this will help me to structure their learning to fill their “toolbox” with strategies that they can hopefully draw on, successfully, and with ease and still fulfil the expectations of their secondary schools.

An interview, as opposed to any other assessment, allows the child to explore and share their thinking in a non-threatening environment. I provide a whiteboard and encourage the learner to share their thinking every step of the way.

I am excited to give our new interview a go and decide how best to provide for my group of children.  I am wondering ….

• Will I attempt to teach the 5 children as a group or will I target them in pairs / individually?
• Will I always be able to provide a “real life” situation? ( I hope so!)
• Will a provocation start each session, no matter how many children? (always my aim)
• Will I be able to successfully move their thinking on to the formal processes, even if the understanding is not 100% in place? (and is this the right thing to do?)
• With the assistance of classroom teachers, giving the children more opportunities to explore and practise their methods and exposure to experiences using them, will these children retain and apply what they learn with me?

I would love some input about my interview from others.

Would this be helpful to you?