Making Meaning

Several years ago, Jo Boloar’s course ‘How to Learn Math’, included a video clip, of Sebastian Thrun speaking about his approach to problem solving and math. I felt a spark of recognition and a sighed with relief to hear him say that we should not move ahead with a problem until we understand it intuitively; that we need to take time to understand and internalize the meaning and context of the problem. Time to think and develop a meaningful context changes everything for learners. Problem solvers become more emotionally vested in a solution when they make connections and link concepts. Every teacher loves the ‘ah ha’ moment when a student thinks, ‘There is sense here!’ and learners who have time to explore, question, and play with the ideas, develop that intuition and agency over their own learning.

Peter Johnston in his excellent book: Choice Words, states:

“There are hidden costs in telling people things.  If a student can figure something out for him-or-herself, explicitly providing the information preempts the student’s opportunity to build a sense of agency and independence.” p.8

And this I love too: ” … most accomplished teachers do not spend a lot of time in telling mode.” p.8

And so, I work to cultivate courage and curiosity in my classroom. Courage to tackle something that is hard, knowing that it is OK to make a mistake, and the curiosity to question what we see and think as we work together. This means that I promote, model and identify those qualities for any challenge we face as learners in my classroom.

I highly recommend, Choice Words by Peter Johnson. I love this book! Can you tell? Perhaps it’s because these ideas fit with my own reflective personal style, but more than that, Peter Johnson excels at demonstrating the power of well chosen words.

There will be more posts to come on this engaging and thoughtful book.

That Moment

Jo Boaler’s course titled: How to Learn Math,  inspired me as a math teacher and learner.  This course challenged me to think of ways to include  Number Talks as part of my online course for grade three students.  One way I have done this, is by developing Three Act Math lessons* in the form of videos to promote discussion between my students and their parents.  As I developed these math conversations in my online course, parents have shared the joys and challenges of participating in this new way of thinking.

Yet I wanted to do more. I wanted to get kids talking and showing what they could do with mathematical ideas, and I wanted kids to see and respond to the thinking of their peers.

There’s even more.  I feel strongly about the power of writing and drawing, as a way to explain math thinking, and so I have students use math journals. However enabling students to respond meaningfully in a math journal is a challenge. Students struggle to reveal understanding when writing skills limit the explanation of their math thinking.

 

Student created math videos is an option I’ve long considered.  And so I started to explore options for easy screen screencasting tools for kids.  I started asking students to recored their math thinking using one of three iPad applications:

  1. ScreenChomp
  2. ShowMe
  3. Explain Everything

My students also have the option of using the video recording tool in Moodle, the Learning Management System I use.  Moodle has a video recording plugin called, PoodLL, (Ha ha of course, you say, what better name could there be?) Happily all of these tools were easy to teach my students to use.

I started by creating my own math video as a model for students.  I used ScreenChomp. Mine, was not polished production but a recording of my thinking and drawing.  My purpose was to get students to focus on the math, and enjoy using new tools. We started with the the following math journal questions from our unit of study at the time:

How can you multiply two numbers?

When so you multiply?

How does an array show multiplication?

Do you ever have that moment when you see or experience something and your skin just tingles with excitement? Well that was my experience as I started viewing the videos my students created.  Not only were they revealing their thinking, the whole process of creating a video powerfully strengthened their learning.  It was evident that creating a math video required my students to communicate mathematical ideas as they explained and supported their reasoning.

As we’ve progressed my students are contributing to a bank of wonderful student explanations of math concepts.  Which in turn, is becoming a rich resource for learning.  I am beginning to think of new and creative ways to use these same videos to develop more math conversations. That’s more to tell in a future post.

Let me know what you have tried to do with students screen casting.

*For many of these ideas I am  indebted to Graham Fletcher who shares 3 act math resources. Find him here: Twitter: @gfletchy  and here: 3-Acts Lessons.

One Common Thread

What a week this has been!  So much to think about.

On Monday Martin Brokenleg engaged us all as he spoke about the Circle of Courage: The spirit of Belonging: I am loved, The spirit of Mastery: I can succeed, The Spirit of Independence: I have the power to make decisions, and the Spirit of Generosity: I have a purpose in my life.  Martin’s words of wisdom, stories and insights as a gave us a deeper understanding of how to connect with youth at risk. Not only youth at risk, but every child that comes into a classroom. He reminded me again of how all of us thrive when we have healthy relationships, the ability to succeed, opportunity to make decisions about things that matter to us and a sense of purpose in all we do.

Later in the week there were conversations with colleagues about, Teaching at the Pace of Learning. This phrase is food for thought.  How is it possible to teach beyond or outside of the pace of learning?  Imagine if a student is not yet ready for the new learning or if a student has mastered the concepts we are teaching? Is it really a teaching and learning relationship then? Or are we both filling in time.

Still, I know it is a challenge, how do we enable teaching at the pace of learning?  So many good ideas were shared in our conversation. Ideas put forward included, refreshers for students at any point in a course, ‘Blue Pencil Cafe’ – a meeting where students mentor each other, providing pace support for students, identifying the critical learning so that a student is ready to tackle the next level successfully, and, identifying the real needs of an individual student, which comes back full circle to Martin Brokenleg’s session on Monday.

One last conversation was about assessment.  Of course, what teacher conversation would be complete without a discussion on assessment? Think the words we use. Whenever I am working away at giving students feedback on assignments I consider that what I am doing is this, ‘supporting student success‘.  My colleague uses the following words which resonated with me, ‘assessment embedded instruction’. Yes! instruction is guided by student needs.  And somehow I feel like this brings us back full circle once again.

 

 

 

 

Thoughts on Student Assessment

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Will this assessment help me to identify student’s needs as learners? Will it help me to guide next steps?  These questions swirl around my brain.  See that determined orange tabby climbing higher to to new levels?  That’s what I want for my students.

Recently this phrase, caught my attention, actionable feedback. Feedback kids know they must act upon, as apposed to feedback that sounds like advice or a mere suggestion. Actionable feedback gives a clear message about the next step or goal.  An example might be asking a student to revise a piece of writing by adding lively action words. It could also be just the right question to push a student’s thinking forward. How will I do this? Specifically identify what was done well, then drive the learning forward with a clear next step or insightful question. This requires mindfulness on my part as I guide students to next steps.

 What about exemplars and rubrics? We have all used exemplars when assessing student work. I have to admit that I sometimes look at a piece of student writing and compare it to exemplars at each level.  Hmmm… is it most like the limited, adequate, proficient or excellent example? I use the exemplars to determine the achievement level.  Now turn this thinking around, the exemplars also clarify the rubric when I assess student work. For example what does ‘descriptive language is simple‘ really mean?  Looking at an exemplar to see how ‘descriptive language is simple‘, is demonstrated, gives me better idea of what that descriptor on the rubric means.  An exemplar should make the meaning of each descriptor on the rubric clear to me and reveal the next step for actionable feedback.

Imagine what this would be like for a student.  How does a rubric and exemplar help a student to self assess? For a student, what does ‘descriptive language is simple’ really mean?  Maybe nothing at all! Exemplars can make next steps clearer for students too; by helping them see what their learning looks like and what is missing in order to move it forward. When a student says my work is like the ‘3’ exemplar, I can ask them why it is not like the’4′ exemplar; this may prompt them to identify a next step and they will be on their way. Actionable feedback once again.

It is about helping students internalize this reflective and iterative process.

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Mathematical Curiosity

Curiosity –  Can you think of a time when you or your kids were mathematically curious?  When it comes to math it seems that we focus on knowledge not on curiosity.

Jo Boaler’s course interviewed several speakers who have a passion for math, people who have what she calls an ‘inquiry relationship’ with math.

Computer scientist, educator and robotics designer, Sebastian Thrun, spoke about having an intuition. I marvelled when Sebastian Thrun explained how he looks at a math problem and develops an intuitive understanding of the solution. In fact he states that we should not move further with the problem until we have an intuitive understanding.  Then he takes a further two weeks to fully solve the problem mathematically. I marvelled because the intuitive understanding of the solution comes first.

Take a look at the qualities identified in an inquiry relationship:

Screen Shot 2014-09-24 at 9.00.48 PMImage from How to Learn Math by Jo Boaler.
  • being curious
  • making connections
  • not worrying about uncertainty or making mistakes
  • using intuition
  • exciting inquiry – you can solve any problem

As an educator I can only say:  “Help me develop these qualities in the young learners in my classroom!”  In fact help me live my life that way.  It would be ever engaging.

My question is: “How do I give students who do not have the inquiry relationship  – this curiosity,  sense of intuition, and connection that makes math come alive?”

Today my online grade 3 students shared their solutions and processes for solving a math question related to patterns. Afterwards we talked about the things that we do as mathematicians to solve a problem. As a collaborative group, I was pleased with the ideas these grade threes identified. However I need to go further to guide my students to as they make the inquiry relationship their own.

At first they may not see how all this happens but as parents and teachers, model and talk about curiosity, courage, intuition and connections  – students will see what an inquiry relationship looks like.  By identifying attitudes, thought process, a willingness to take risks, and communicating that math is an engaging challenge and fun, I think we can guide students to develop this passion and connection.

There is more.  Intuition and curiosity are linked to understanding and confidence. Jo Boaler describes these qualities as a double helix.

  Understanding and Intuition

Confidence and Curiosity

These attributes are iterative, a child must develop the understanding to gain intuition and intuition carries understanding further.  Confidence grows as curiosity is satisfied and curiosity depends on the confidence to explore.

 Cultivating those qualities in a math classroom means that as a teacher I  promote, model and identify those qualities as we engage in a mathematical world.

Teacher Language Matters

As I listened to examples of number talks in the math classroom, this stands out, teacher language matters.

“I think I heard you say.”

“How did you know you should have…..”

“Where do you think your mistake came from?”

“So you are saying…”

“How did you figure that out?”

“Do we see it another way?”

“How do we see this one?”

The teacher’s language conveys that effort, thinking processes  and grappling with the ideas are what matters. Students have the freedom to explore the ideas.  Relational equity develops as each student’s contribution is valued and analysed in an effort to come to a common conclusion.

And there is more … students see that math is a beautiful, creative and connected subject.

 

 

 

Creating Safe Places to Take Risks for Learning

What messages will you give your students about the value of mistakes as we learn?

Mistakes, fertile ground for growth.

Live on the learning edge – don’t be afraid to make a mistake.

The path of learning is littered with mistakes – thank goodness!

Mistakes are the stepping stones to learning.

If you can’t make a mistake you can’t make anything. 

How to grow new pathways in your brain:  

1. Take on a tricky problem.

2. Never fear mistakes.

3. Share your thinking.

4. Persist till you meet success.  

Parents and teachers, what messages do you give your kids to encourage risk taking as they learn?

 

Taking on the Challenge

And the moral of the story is….

Ugh!  A didactic tale can fall flat. With little meaningful connection to the story the message is seldom remembered or lost to the reader.  You may identify with this.  I for one, recognize this didactic interaction not only in books but in my classroom.

Lesson 4  of How to Learn Math with Jo Boaler  describes the didactic contract in the classroom.  See if you recognize it.

The Didactic Contract  identified by by Guy Brouseau, states that there are certain expectations for both the student and the teacher in a learning setting.  Teachers are to demonstrate and guide their pupils, and students are to learn with ease.

Here is how it happens in my classroom.  During a math class I quickly step in to clarify, add to a student’s math explanation, or demonstrate the next step. Students on the other hand, are quick to ask for help  when faced with uncertainty because they do not necessarily believe that learning involves struggles or challenges. Together we fall into this unspoken contract. This way of interacting in the classroom becomes a barrier to learning.

Jo Boaler describes it well: We empty the interaction of learning and reduce the cognitive demand for the student.

As a teacher responsible for student learning, it is too easy to take ownership of something that belongs to the learners. Instead of this didactic interaction I need to allow mistakes, reflection, redirection, and meaning making, as students develop math skills.  My goal as a teacher is to have an engaging classroom where students  pursue questions and discuss their thinking about mathematical processes.  This dynamic interaction would promote deeper thinking and meaningful problem solving.

It is critical in this setting to ask the kinds of questions that promote meaningful engagement.  What is a good math question? Here is a challenge I love!  I am building on what I know and will share more with you in a future post.

 

Shout it from the Roof Tops!

Current research helps us understand what high achieving math students do and it is most interesting, in fact Jo Boaler tells us it is worth shouting from the rooftops so here it is:

High achieving math students use flexible thinking,  are able to easily decompose and recompose numbers and naturally compress ideas to move on to harder concepts.

What does all this mean? Let’s look her explanation.

Consider a simple computation such as this:

 5 + 14

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There are several possible strategies to use.

Counting all: First count all the blue counters to 5.  Next, count the purple counters to confirm that there are indeed 14.  Lastly, proceed to count all the counters from 1 to 19.

Counting on: Count the first set of blue counters to 5 and count on to 19.

Known facts: A student may simply know that 5 + 14 equals 19.

Derived facts: Students use what is known about numbers and facts to complete the computation.  Fourteen is also 10 + 4 so, since 4 + 5 equals 9, add 10 and solve the problem to get 19.  Although this may seem obvious to an adult reader it is important to note the process of decomposing and recomposing the number to make the problem easier.

People who are good at mathematics decompose and recompose numbers all the time.” Jo Baoler What’s Math Got to Do with It? p.148

Interestingly enough students who are low achieving at math use approaches that are more difficult.  For example, imagine using a counting back strategy for subtraction.

25 – 14 = ___.

There are many steps involved in counting back from 25.  This is a complex task and one where a student can easily become confused.  Students who find math difficult often apply a ‘follow the rules’,  problem solving process, lacking the understanding to make sense of numbers in flexible ways.   How much easier this very problem would be, if the student worked flexibly with numbers as was done in the earlier example. Fourteen becomes ten and four. Now the problem becomes  25 – 10 = 15 and step two,  15-4 = 11.

Open Cones or Long Ladders? 

High achieving students compress mathematical ideas.  What does this mean?  Think about learning multiplication. Initially students struggle, work through the process and practice examples.  Once students understand what multiplication is, and how to use it, the concept is compressed and easily used in new settings.

coneladder

Jo Boaler  uses an image of an inverted cone to show what is meant by compression and how it helps students as they learn.   Learning is compressed as students begin to apply understanding efficiently. New learning is built on compressed ideas and  understanding grows.

Low achieving students who work at trying to remember rules, methods and procedures have a different view of mathematics, much like an endless ladder to be climbed and a long series of steps to be remembered. These students need to be guided to develop the skills of working flexibly with numbers, and to develop a deeper understanding of number sense.

This post is a summary of the information presented in Jo Boaler’s book: What’s Math Got To Do With It?  Key Strategies and Ways of Working Chapter 7.  Also available at the St. Albert Library.

 

Your Brain is Amazing!

All I can say is, “Thankfully we are developing a deeper understanding of how we learn.”

Once upon  a time I lived in a world of the fixed mindset¹ and likely you did too. Some students were smart and others were not.  Intelligence was viewed as fixed at birth and one of the roles of education was to sort students and direct each one to the correct vocation.   And,  if you think about it, too often we still organize learning in this way.

Let’s identify the qualities this fixed mindset promotes and you will recognize it right away.

Students working under a fixed mindset:

  • Are afraid to make mistakes
  • Avoid challenges
  • Give up more easily
  • Fear of constructive criticism
  • Feel threatened by the success of others

Think what this does to learning. Consider what happens when students see themselves in this way.

Contrast this with a growth mindset¹, a term which on its own sounds encouraging. A growth mindset states that intelligence can be developed and our true potential is unknown.

Students with a growth mindset are:

  • Persistent
  • Not afraid of mistakes
  • Willing to take on a challenge
  • Resilient
  • Inspired by the success of others

How can these ideas change instruction?

It is not the student who ‘knows’, that we should recognize rather the student who says, ‘hmmm I am trying to figure this out and I have not got it yet. It is this student who demonstrates a growth mindset. As educators we need to communicate that everyone can get better if they work on it, which means that persistence becomes a  key quality to encourage.

And for this reason our view of mistakes plays a critical role in our mindset. How do mistakes impact learning? The student who makes a mistake has multiple opportunities to learn.  First from recognizing the mistake, and then from working through the process to correct it.  All this creates more opportunities for brain synapses to fire and grow. Working through mistakes causes our brains change and develop.

What message shall we give to students?

 Mistakes are fertile ground for learning.

As an educator with a growth mindset, I am motivated to create an environment where risk taking is safe and encouraged, and where  learners at all levels are recognized for their effort.

 

¹Carol Dweck – Mindset: The New Psychology of Success Random House Publishing Group December 26, 2007