Number Flexibility, Mathematical Reasoning, and Connections

This is the fourth lesson in Jo Boaler’s “How to Learn Math (for Students)”.

She starts by discussing some more math myths:  That math is just a lot of rules that need to be memorized, that it is all about right or wrong answers, and that it is not a creative subject.  But studies show that high achievers think of numbers flexibly and low achievers believe there is only one correct way to do any problem.

For example, there are many ways to work out 18 x 5:

Multiply 10 x 5 = 50 and 8 x 5 = 40 then add the results, so 50 + 40 = 90

Multiply 20 x 5 = 100 and then subtract 2 x 5 = 10, so 100 – 10 =  90.

Since 5 = 2+2+1, multiply 18 x 2 = 36 and double that 36 + 36 = 72 and then add the last 18, so 72 + 18 =90

18 x 5 is the same as 9 x10 because 18 = 9 x 2 so 18 x 5 = 9 x 2 x 5 and 2 x 5 is 10, so 9 x 10 = 90

That was four different methods – there may be more!

But some students believe that it is not allowed to change numbers in this way, but this type of flexible thinking is really important. Students can get in the habit of thinking about multiple ways to solve any problem.  One way to develop this habit is to work with others.

Math is not an individual, solitary activity. Talking about math can greatly increase a student’s level of understanding.  In fact, mathematician Uri Treisman studied UC Berkeley students who were failing calculus classes and leaving the university. He found that there was only one difference between students who were failing and those that were not:  The successful students worked together on their math and talked about it, where the unsuccessful students worked alone. So Treisman put the failing students in study groups, and within a year they started out-performing the other students.  It works because discussing problems engages students in reasoning and metacognition.

Lastly, it is important to remember that all math is connected.  It is not just discrete topics that need to be mastered. For example, proportions come up again and again in math over a lifetime of math learning – fractions, similar triangles, dilations, graphs, slope, rates of change, the Pythagorean Theorem and Pythagorean triples.  Higher achieving students see these connections to big ideas, whereas lower achieving students just try to memorize methods.

The PISA study showed that students with a fixed mindset (who believed that people are either good at math or not) achieved at lower levels, and the kids with a growth mindset were the highest achieving. And students with a growth mindset who also used positive strategies (who made connections and didn’t memorize) were the highest achieving students in the world.



And away we go!  First week back at school.  I know, my district starts comparatively late, on the traditional day after Labor Day.  (Fun aside:  I just found out the reason why so many districts now start in August.  Standardized Testing.  The tests are in mid-May, so many districts figured, why teach after that? So starting in August gave them additional time before the tests.  Then their school year is done as soon as the tests are finished.)

This week, I tried the “First Day Graffiti” project. Only I did it on the second day of school.  Because I’m a rebel like that.

My topics were:

It makes it EASIER to learn in a class when…

It makes it HARDER to learn in a class when…

I am most likely to participate in classes when…

Other students help me learn when they…

I learn best when the teacher does…

Here are my results (click on the picture for a larger view):

2014-09-03 15.44.50 2014-09-03 15.45.05 2014-09-03 15.45.16 2014-09-03 15.45.25 2014-09-03 15.45.34 2014-09-03 15.45.56

In a nutshell, students want:

A quiet classroom

Instruction at a slower pace

Group projects

Teachers with a positive attitude and mood

Help from their friends

Clear, step-by-step instruction with lots of examples

I’ve posted these in my classroom as a reminder when I plan my lessons. These students really care about their education, and they know what they need.  I need to listen.


This is the third lesson in Jo Boaler’s “How to Learn Math (for Students)”.

Our brains grow in response to stimulus, and mistakes are stimulating!  When we get an answer correct, there is not the struggle needed for brain growth. So it is important to try difficult problems and make mistakes in order to build a strong brain.

As Thomas Edison said, “Genius is one percent inspiration, and ninety-nine percent perspiration.”

The more mistakes we make, the more successful we are.  So how do get this mistake-making experience?  By trying different things, being okay with being wrongs, and by not judging ourselves or our ideas.

As we move in to Common Core, students may be frustrated by the open-ended nature of the problems they will begin to encounter.  It is important to keep a growth mindset and be determined to struggle on, knowing that mistakes actually aid the learning process.

Fast math isn’t always good math.  Math requires deep thought, and deep thought takes time.  What is essential is making connections and asking (and answering!) questions such as why things work the way they do.