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.

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




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.


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