# 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:

###### Image 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.

# My Mother and ‘The Number Talk’

My brother nodded in agreement and chuckled, “Yes our mother was great with numbers.”  She had the ability to use numbers fluently, a skill we we both easily acknowledged.

How well I remember standing next to my mother in a grocery store or bank as she quickly and easily calculated totals.  I would line up the digits mentally in my head, carry or borrow if needed, and try to remember the numbers as I worked.  What a cumbersome procedure!  She, on the other hand, already had the answer and cheerfully let the clerk know the amount required.  I recognized her skill and longed to have it.  It wasn’t speed I wanted, it was the ease with which she worked. Why couldn’t I do that?

Was it the education my mother received during the 1920’s in the Netherlands?  Or was it as a young, single woman when she worked for Unilever that she developed proficient number sense? It was not until later in my mathematical life that I learned her secret. Perhaps I would have benefited from a motherly ‘Number Talk‘.

A Number Talk?   Yes, the kind of talks I plan to use with students this year.  Parents certainly can engage their children in number talks. They’re fun!  And teachers of math will find them a best practice for developing number sense in students.

I observed a number talk as part of my summer math course. Here are my observations.

How Number Talks Work

The instructor sets the students up for the number talk telling them they are to figure out a math question.

1.The question is written on the board and students are asked to solve it. No pencils, no paper, they are to calculate their answers mentally. No comment is given on how to approach the problem.

2. Students are given lots of thinking time and respond with a thumbs up once they have the solution. A low key way to respond so that everyone can take the time they need.

3. Students then share their answers and explain how they arrived at the solution.

4. Through the discussion students are led to see the interesting variety of approaches used to solve the question.

Open ended questions such as these spur the discussion on.

“Anyone try something different?”, “Anyone else do that?” These questions give everyone a chance to explain their process.

“I think I heard you say.” or “How did you know you should have…..” These statements give the student opportunity to clarify thinking and communication. The teacher does not add to the student’s explanation she only repeats what she hears. The onus is on the student to ensure that the explanation is clear.

“Where do you think your mistake came from?” Helps the student clarify logic and identify the error. Mistakes are part of the solution process both acceptable and interesting as part of the learning.

5. The process students use to solve the problem is written on the board so that it is easy for everyone to follow. This is important. Everyone needs to understand how the numbers are manipulated.

6. All solutions are represented on the board and students are asked to draw a picture of their solution and someone else’s solution. Another important step as it makes the learning more concrete and helps students see how they can work flexibly with numbers.

Screen shot from XEDUC115N How to Learn Math

Number talks are one way to help students develop insight, ability and willingness to to break numbers apart and regroup them as they observe and discuss different ways to solve math problems.

Yes, the answer is important but it is not the most interesting part of a mathematical question.  As we show students that problems can be solved in different ways we teach them the very, very important building block of number sense, a skill that is foundational for the rest of math.

Ahhh yes,  a number talk would  have given me insight into the skillful mind of my mathematician mother but I think I am on to her secret now.

# 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.

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.

# 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.