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Articles & Seminars
The biggest problem with the teaching of mathematics today is that it can be boring
and have little meaning or interest for students. If you look at any maths book available for school children, the overwhelming thought would be dreary, dull, repetitious, and well, just plain hard work. Compounded with this is the problem that most of the population will tell you that they hated maths at school and are no good at it even now.
Maths does have a definite image problem.
So what can we as teachers do about this, how can we encourage our students to have positive attitudes to maths and to think mathematically?
MATHEMATICAL GOSSIP of course.
Human beings are very good at gossip, it helps people in a group get to know each other, to find out intimate details and develop an understanding of other individuals. As soon as we know a little bit about someone our interest is aroused, we want to know a bit more and these understandings lead to an interest in one another and ultimately individuals in a group begin to care about each another.
THE BIG SECRET is that people who are good at maths are interested in it, they care about it, and they want to do it. Mathematicians 'gossip', or chat about numbers, and this is how their interest in things mathematical develops, our students can begin to develop this interest in just the same way.
Keith Devlin thesis in his wonderfully titled book, "The Math Gene - How Mathematical Thinking Evolved and Why Numbers are like Gossip", is "that thinking mathematically is just a specialised form of using our language facility. That mathematical ability is innate just like the innate ability to acquire and use language, in fact it is just a more specialised version of language. So here we have it in a nutshell maths is like gossip, maths is language based, maths is discussion, sharing and developing of ideas and hypothesis. This is because maths is not just about numbers, in fact arithmetic is just a small part of the study of mathematics, it is about ideas, relationships, abstract thinking, cause and effect, logical reasoning as well as algorithms and numerical ability.
Real mathematical thinking comes from considering ideas and relationships, from logic, from hypothesising, and these ideas need to be developed using language. Language, discussion and collaboration give maths its meaning.
As with reading there are fundamental building blocks for mathematics, things like counting, addition, and multiplication facts. But even these are really more interesting than phonics for heavens sake. Pythagoras (565-545BC) would have been over the moon to have had the knowledge that any average primary school child has today. In the BC days, maths was like the internet of today, there were wizards, magicians, and the truly gifted and talented. There was a great buzz and excitement over new discoveries. People meet to discuss their latest maths ideas, and in fact this is still happening.
Paul Erdos was one of the most important mathematicians in the twentieth century.
"For more than 50 years, Erdos wandered the globe visiting mathematicians, attending meetings, teaching, and lecturing. He had become the centre of an enormous web of collaboration."
It is through collaboration that the sense of discovery, the fun and excitement of mathematics can be found in the early learning years of mathematics. Collaboration teacher to teacher, student to teacher and student to student.
The rewards of effective discourse are many. Not long ago, I was observing students work out a problem together. When they finally "got" it, one of the students exclaimed,
"We are so smart."
Isn't that what we are striving for? Not "the teacher is so smart," not "they are so smart," but "we are." In that statement, I heard both confidence and community. Healthy discourse promotes these things. -- Cynthia Lanius, Project teacher
What questions do you ask, how do you stimulate a task discussion, how do encourage mathematical thinking?
The Maths Forum, above offers lots of splendid ideas for encouraging mathematical thinking, including various approaches, leading and non-leading questions, paraphrasing summarizing, listening and assisting with decision making. There is even a video clip demonstration in the use of excellent questioning techniques. But wait there's more, this site offers some excellent lesson plans for problems that stimulate mathematical thinking.
The internet is a grrrrreat place to start discovering mathematical problems. Working in groups students can have great fun 'gossiping' about maths, working out hypothesis, proving their theories and inventing solutions.
To help with these discussions a method of attack is useful. Here's one I often Use:-
Stage 1 - Brainstorm
1. How are we going to work this out?
2. What do we know?
3. What do we need to find out?
4. What strategies can we use to solve this problem.
Stage 2 - Solution
1. Use strategies discussed.
2. Discuss various outcomes.
3. Modify strategies if necessary.
4. Find a solution, and give reasons for your solution.
Stage 3 -Reflection
1. How well did we solve the problem?
2. Did the group keep on task?
3. Did everyone contribute?
4. How did you feel about your final answer?
From the 'Centre for Innovation in Teaching' comes some challenging problems you might like to try. Here's one to wet your appetite:-
"When one particular number, written in figures, is turned upside down it increases in value by 21. Which number is that?"
All the problems at this site require planning and discussion to solve and a method of ATTACK. When you arrive at the site, click on "Puzzles, Pastimes and Competitions". The Quick Puzzles are in the first set.
Games to develop Mathematical Thinking -for the Wizards
Below are listed some links to mathematical games. These games are best played in groups of 2 or more, and students can discuss strategies, and ideas in groups as well as with the whole class.
Matchstick Puzzles, Word Ladder Puzzles, Kicks Try them all
Mathematical Food for the Mind - For the Truly Gifted
Enter a number and I'll tell you everything you wanted to know about it but were afraid to ask.
Famous Problems in the History of Mathematics
This site will be of particular interest to gifted and talented students. It covers the "The Bridges of Konigsberg", "The Value of Pi", and many more, in simple language.
Create a mathematics buzz in your classroom. Celebrate original thinking and new discoveries. Go beyond the multiplication tables and number facts and into the world of Mathematical Thinking. Make friends of numbers, talk about them give them personalities, make jokes about them. Talk and gossip about them with your students. Build a mathematical world that everyone is interested in and wants to be a part of.
Take a Look at the VT Maths Page for more ideas.
Oh! here's one to get you started.....
No, you need multi-plier?
Keith Devlin (2000) The Math Gene - How Mathematical Thinking Evolved and Why Numbers are like Gossip Weidenfield and Nicolson Great Britian
Paul Strathern (1997) The Big Idea Pythagoras and his Theorem Arrow Books United Kingdom
Paul Hoffman (1998) The Man Who Loved Only Numbers Hyperion NY USA
Cathy Brown (2000) Do you need a Hammer to do Multiplication
Cathy Brown B.Ed. (HONS) is a Educator with over 20 years teaching experience from Pre-school to Tertiary level, she gives seminars and inservice in Computer Technology and also writes a free email newsletter Virtual Teacher Read it at http://www.virtualteacher.com.au