Thinking Mathematically

I got an e-mail a few days ago from a friend. It contained one of those typical requests to change the world by sending on my e-mail in the following way:

I am sending this note to a lot of people. If all of you send it to at least ten more (30 x 10 = 300)….and those 300 send it to at least ten more (300 x 10 = 3,000) … and so on.  By the time the message reaches the sixth generation of people, we will have reached over THREE MILLION consumers! If those three million get excited and pass this on to ten friends each, then 30 million people will have been contacted! If it goes one level further, you guessed it…..

THREE HUNDRED MILLION PEOPLE!!!

Now the way that I was educated in mathematics at school was the old-fashioned listen to the teacher who gives you the way to get it right. On that basis I would follow the logic of the above statement and say yes… it is all about multiplication and I can multiply by 10 so it must be accurate… wow imagine getting to all those people in just a few generations of e-mails!

But I have re-educated myself as a teacher and now as a mathematics consultant. I have come to realise that the old certainties do not exist in the real world. I was not taught to think but just to act and get a result… thus every triangle has 180 degrees in it and so if I know two angles I can work out the third. The formula for the area of a triangle is half the base times the height.. if I know these two measurements (or can deduce them) then I can get the answer.

In the real world there are uncertainties. There is no value in having a mean average of  26.5 in the size of classes in a school.. where is the half  child?  There is no exact measurement of the height of a building,  my measurement is accurate to within a certain fraction of a centimetre but cannot be exact. How many people are likely to enter the city of London today? How do I know? There are so many factors that can affect this number (too many to list).

I have spent a lot of the weekend reading an excellent book on mathematics called “The Elephant In The Classroom” by Jo Boaler. This book tackles the reasons why children often hate mathematics and find it difficult. The sub-title of the book is “Helping  Children Learn and Love Maths” (The book is published in North America as:  “What’s Math Got to Do with It?: Helping Children Learn to Love Their Most Hated Subject–And Why It’s Important for America”).

The central idea of the book is that the old way that I was taught does not allow children to really enjoy or experience mathematics, it is about certainties and answers and ticks on the page. Boaler puts forward the idea that mathematics should be about exploration, collaboration, problem-solving and playing around in search of the patterns that excite mathematicians but which is mostly missing from many school’s mathematics teaching.

Looking at the generating e-mails example above with the eyes of one who does not expect certainty I can say that, in the real world, there are networks that overlap and therefore the simple progression towards 300 million will not happen… thirty people may well send the same e-mail to the same person (probably me!) and I will send it on to some hapless person who will have received maybe one hundred of these and in the end will just delete it through frustration!

In Boaler’s book she gives an example of two schools that she has looked at in a longitudinal study. One school, Pheonix Park (made up name) promotes a collaborative, problem-solving, mixed ability type of teaching in mathematics whilst the other, Amber Hill, promotes a traditional “sit in rows and learn the rules” type of approach. She contacted students from these two schools years after they had left the schools and were now adults out in the workforce. She found that the ex-students from Pheonix Park had more professional positions and pay than their counterparts in Amber Hill although in socio-economic terms Amber Hill was the better off school.

What was most interesting was to read the quotes from the ex-students from the two schools. In relation to examinations and tests the two different approaches produced different attitudes from the students. Thus:

“The Pheonix Park students had not met all of the methods needed in the examination, but they had been taught to solve problems and they approached the examination questions in the same flexible way that they approached their projects – choosing, adapting and applying the methods they had learned.” (Boaler, 2009, p.62)

Trevor, a student from Amber Hill said the following when he was asked about tackling exam questions. He said, ” You can get a trigger, when she says like “simultaneous equations” and “graphs” or “graphically”. When they say like – and you know, it pushes that trigger, tells you what to do.” I asked him “What happens in the exam when you you haven’t got that?” he gave a clear answer “you panic.”

I went to a school like Amber Hill but I would only teach in a school such as Pheonix Park if I had the chance to choose. The Pheonix Park student may well have noticed the mathematical problems with the “generating e-mails” idea. Maybe the Amber Hill student may have noticed but there is a good chance that he/she wouldn’t have. In an employment situation we surely need people who can think, question and discuss possibilities and not look for “triggers” that give them answers!

It still seems strange to me that at a time when we need to be promoting problem solving and collaborative skills we need to be looking at books such as “The Elephant In The Classroom” and following the advice of allowing children to enjoy mathematics by problem solving together, recording their results and discussing ideas and not going the other way of promoting the “chalk and talk” and certainty. We should not be rigidly setting our children and labelling many of them as hopeless at mathematics. We should be promoting a generation that is flexible and adaptive and does not seek “triggers”!

 

 

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