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

It’s a scary mathematical world out there! Hmm… really?

Math, math, math, what are your thoughts on this subject?

Is it true that math is portrayed as a hard subject? As a student have you ever received the message that  some people are math people and others are not?

Do we hold stereotypical messages about gender or race and ability to do math?

When you were in school what did you think about your own ability to do math?

You might be surprised to hear:

“All students can achieve at the highest levels in maths at all levels of school right up to the end of high school.”

Yes, there are countries in the world where this is the expected norm.

This summer I am using this blog to reflect on my learning in the course: How To Learn Math by Jo Boaler. This course is intended for teachers and parents and presents new research ideas on learning, the brain, and math that can change the way you think about math and how we learn.

The ideas on this blog will be a combination of my reflections and notes from the course.  My hope is that along the way I’ll add clarity, and a deeper understanding to what I already know about math instruction and gain new ideas on how enlarge and enrich the world of math for my students. I hope you’ll join me in this adventure.

 

 

 

An Online Math Course verses Summer Reading

This summer I am taking an online math course from Jo Boaler, Professor of Mathematics Education, Stanford University.

It is fitting I think, that an online math teacher should take an online math course and it is logical then, that one of my interests is the very structure and organization of the course. How do I as a student interact with the course and with other students?  In what way is this course engaging? How do I assess my progress?  Does the course provide resources for further learning? And how on earth can an online course compete with summer in Alberta?

Summer reads verses online math courses, its a tough competition.

Summer reads verses online math courses, its a tough competition.

It must be engaging, and you may be happy to know it is, both in content and structure, because for a period of time each day I am passing up on warm sunshine and relaxing beach reading, to sit in front of my computer.

What are the features of this course that make it work for me? Each lesson consists of a series of  short videos with accompanying text.  I can view the videos several times if I wish and stop at any point to jot notes. The videos are short, from less then a minute to about 12 minutes and each video ends with a question that asks a response from me. In addition each video includes a forum where participants can reflect and comment on content. As a whole the course is easy to navigate, I can see my progress and understand what is next.

Already I can see these strengths:

– short chunks for learning

– immediate opportunity for response to the content

–  interaction with other students

– flexibility to work through the material in any order.

– easy to navigate

– learn when I want ( and enjoy the sunshine too!)

Have you ever taken an online course?  How was it structured? Did the course format work for you? What worked well?  What would have improved the course for you?  I would like to hear about it.

English Language Learners in the Czech Republic

As I look back to the summer of 2013, countless memories remain from my experience as an English language teacher in the Czech Republic.

The beauty of Prague? Yes surely.
The challenge of daily communication? That too.
The warmth of people in the Czech Republic ? Absolutely!

All this and more….

What a crew we were, several accomplished ELL instructors and a group of enthusiastic, willing, professionals prepared to share part of their summer with English Language Learners in the Czech Republic. We were motivated by faith, striving in a meaningful way to share the life of faith in Christ, by leading a week long English Camp.

English Camp

Could all the hours of preparation prepare us for the unknowns our team might experience as we conducted our week long English camp? Hmmmmm….. likely not.

Lessons for several levels of learners had been prepared with care, daily welcome and ice breaker activities were ready to go, evening programs to close the day with energy and good spirits were included in our arsenal of plans.

In the Czech Republic, students age 12 to 25 and several adults aged 55 and 75 had set aside a week of daily life to be learners. They were an eclectic and amazing bunch, from all walks of life. Our classes included adults who had experienced the Russian occupation and young people who were eager for a new future. As I think of them now, a year later, they were role models of students eager to learn.  They were persistent, willing to learn from mistakes, and encouraged by each other’s success.

As a teacher I was happily immersed in the joy of learning for one short week with these lovely people.   These students demonstrated that ability and ‘smartness’, grows with experience.  These English language learners had a growth mindset when it came to developing new skills.

Learning involves risk taking and all of us, teachers and students, had to be willing to jump in and try something new. Each of us would evaluate our attempts at communication, revise, laugh to ourselves and forge ahead. How I applaud these students, who were willing to take the plunge. Sometimes it was with shyly spoken words and other times with full expression in a reader’s theatre.

Shared meals and conversation developed a learning community, as we laughed, cheered, problem solved, and persisted to share stories about ourselves. How could I forget, warm and gentle Pavell who came to learn this year because he heard about the class from others. Or wise and kindly George who approached my husband, Sid, on our last day, put his hand on his own heart, then reached out and put his hand on Sid’s heart, as he quietly said, “We are brothers. ” We reached beyond language barriers to truly communicate.

Our work was built on careful organization by our host church , First Baptist Church of Litomaurice. Every last detail, food, lodging, teaching facilities, and student registration was carefully thought out and arranged. How thankful we were for the love, care and commitment that went into this preparation.

We made connections across the globe and as I reflect on last summer I see no distinction between learners and teachers, we all came away changed.