• Contact

  jannordgreen@gmail.com

• Categories

  • A2 level
  • AS level
  • Calculator
  • Challenging the students to come up with questions for them to answer.
  • Functions
  • Game theory
  • Humour
  • IGCSE
  • Learning
  • Note
  • P2PU Course: Create and Share Math Interactives
  • P2PU Course: Mathematics Curriculum Development
  • Problem solving
  • Programming
  • Software
  • Teaching
  • Textbooks
  • Video

• Archives

  • May 2011
  • April 2011
  • March 2011
  • February 2011
  • January 2011
  • August 2010
  • July 2010
  • September 2009
  • August 2009
• Search for:

• Blogroll

  • Math blogs

• Links

  • Diigo
  • thnik again!

• Meta

  • Log in
  • Entries RSS
  • Comments RSS
  • WordPress.org

There are two kinds of shears.

1. Horisontal shear.

All points are moved horisontally, i.e. the y-value does not change. How much a point is moved horisontally is proportional to its distance from the x-axis.

In short: (x, y) -> (x + ky, y).

k is called the shear factor. The x-axis is called the invariant line.

2. Vertical shear

All points are moved vertically, i.e. the x-value do not change. How much a point is moved vertically is proportional to its distance from the y-axis.

In short: (x, y) -> (x, y + kx).

k is called the shear factor. The y-axis is called the invariant line.

Question:
What kind of shear maps triangle X to triangle S? What is the invariant line? What is the shear factor?

The problem:

James Tanton has made a video where he eats a piece of a cake. The white square below.

Carefully rearranging the pieces that are left he manages somehow to make the cake whole again!

Voila!

Please explain how this is possible.

 

Khan interviewed by Charlie Rose last week.

On the usefulness of Khan’s videos.

“[what is needed for a good mathematical task is a] clear premise in its first act, obstacles, conflict, and tension for your classroom heroes to resolve in its second act, and a cathartic resolution in its third act that leads naturally and necessarily to more mathematics in its sequel.” (more)

The quote is taken from Dan Meyer’s blog. The words in brackets I added.

Whenever I use a word Jan Thomas (5 years) doesn’t understand, he will ask. Yesterday, he asked “What does ‘pattern’ mean?”.

My answer, which I don’t think was very successful, said something about not being random, having a rule, a system, pointing to the stone pattern in the pavement we were walking on.

Today, in New York Times, I found Leanne Shapton’s “Wednesday’s Patterns” I will show him today. The pattern in the image above is from ‘a towel left behind by house guests.’

Belief 1: School should be life itself, not the preparation for it.

Belief 2: The task of the teacher is to create a learning environment.

Belief 3: ‘Modern mathematics teaching should start with problems and attempts at solving them, which should lead to a mathematics as a strategy of such attempts.’ (Erich Wittman)

Belief 4: Students learn best when they feel ownership of their education, when they learn things that matter to them, and when they learn together.

Belief 5: We need to meet kids where they are. Not where we wish they were.

These are some of my current beliefs.

In steps reality:

The chapter on matrices and transformations, in the textbook I teach from, introduces matrices out of a hat. The students are told to practice how to add, multiply, and find their inverse before anything is said of why matrices may be useful. When matrices are used to represent transformations it is kept as a secret why this approach may be useful.

The challenge:

Write a resource book (text book is a word with too many bad connotations) in harmony with the believes above that, among more important things, prepares them for the external IGCSE exam.

 

 

• teacher –  someone who tries to create a learning environment (my definition)
• a person who teaches or instructs, especially as a profession; instructor (source)
• A person who cares enough about abusive and ungrateful teens to work for crappy pay and long hours while hoping someday students mature … (source)

 

The other day I discussed this question with a student of mine. I stressed that a mathematician chooses his problems. He chooses them based on his interests, what he think he can accomplish, and is free to drop the problem or change into another if he so pleases. I also stressed that no mathematicians work in a vacuum, that they need discussion, results, inspiration from other mathematicians, and to know their solution is valid they need the stamp from other experts in their field of study.

The above is illustrated very well in the story of how Andrew Wiles cracked Fermat’s last theorem.

Task 4 Week 1 in Mathematics Curriculum Development:

Find someone with a strong voice talking about a vision of mathematics education, and a place where people are commenting on it, such as a blog post, a forum topic, a Ning or LinkedIn discussion, an Amazon.com book review collection… Contribute your critique of the piece to the discussion, based on your own vision of math ed (Task 3). Before posting to the discussion, post your writing to your blog, both to grow the blog and to make sure your content is preserved against posting accidents, deletion by moderators and other causes of loss. If you are moved to critique more than one piece, use this form multiple times.

One of the comments to this video on YouTube:

Which four countries educate their kids best at math? Answer: Singapore, Korea,Japan and Switzerland. All these countries apply a real world problem solving approach to maths lessons as an initial starter for all lessons then they figure out the maths needed to solve the problem.

I don’t know if the answer is correct, but I believe that all lessons should start with a problem.

Erich Wittman put it like this:

Modern mathematics teaching should start with problems and attempts at solving them, which should lead to a mathematics as a strategy of such attempts.

To me the most pressing problem is to make the learning of math meaningful. Two steps are required: 1. start with problems and 2. let the computer carry out the algorithms while we concentrate on which algorithms to use and how to interpret the results.

Right now I teach math to AS and A2 level setudents. It is truly ridiculous how much time and energy we use to learn and carry out algorithms that http://www.wolframalpha.com/ does in a second. (Joke: I actually use wolframalpha to get hints on how doing things manually using the Show Steps facility).

There are of course excellent free programs that can do the tedious work for us. One is wxMaxima.

That computers should be used in exams is a simple idea that should be carried out, but how many years will it take before it is realised? Another option is of course to abolish the exams and let the teacher evaluate the student.

When step 1 and 2 are in place we are ready for step 3: let the students find the problems to study.

This was the request:

Make a blog if you do not have one, or use your existing blog. Write a post introducing yourself to other course members. In the post, also describe your dreams and plans for mathematics education, in the context of curriculum development – the topic of this course. You can select any scale: imagine yourself the national or world czar of math ed, or make plans for your community, your class, or for your family. Put the url of the blog post in this form.

A brief introduction of myself is given here.

Instead of dreams and plans I like to mention my most pressing problem.

I currently teach mathematics to 15-19 year olds at a private school in Tenerife, Spain. The students take external exams from Cambridge, England.

Where do I find a textbook for IGCSE that starts each chapter with a problem that is meaningful for the students and give the students ample time to solve it with their teacher?

Here are a few more pages on the problem:

• http://mumnet.easyquestion.net/action/action001.htm
• http://www.easyquestion.net/mumnet/guests/guest001.htm

Next Page →