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.


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