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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.

There are seven tasks:

Task 1. Design an algebra activity…

Five sub tasks are listed. One of them:  ”include features that make the activity meaningful for students.”

I return to Erich Wittman:

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

In other words, you don’t start with ‘today I want to learn them second degree equations and I’ll find some really nice applications to show how useful it is.’ Instead you discuss with the students what concerns them. The day they can’t think of any, you may ask them:

What causes most deaths amongst children between 3 and 15 years of age in the United States? If you don’t know the answer, write down three guesses.

What happens then depend on their answers. Here are some possibilities.

Task 3 was an invitation to edit a Wikipedia article on mathematics education. I added this paragraph to this page:

Lakatos argues for a different kind of textbooks, one that uses heuristic style. To the critics that say they would be too long, he answers: ‘The answer to this pedestrian argument is: let us try.’

To me Wikipedia is an excellent aid in studying mathematics for its breadth and depth of articles. I used my name as alias so it is easy to find me when you click ‘View history’ in the Wikipedia page.

Task 4 was about using WolframAlpha. Sadly enough the students in Alevel math or IB math are using most of their time and energy on carrying out algorithms that WolframAlpha or the free software wxMaxima can do faster and more accurate. If they could use these tools they would have more time tasks the machines are not good at: finding problems, formulate them, ask questions, and interpret answers.

Here is a simple task WolframAlpha does in a snap: finding partial fractions. An example.

Task 6: “Attend a live event at one of online mathematics education communities. “

I attended a presentation on Wolfram Education with participants from all over the world.  A Wolfram guy talked about the how statistical data can be presented in Wikipedia 8. It was one-way communication for the first 20 minutes so I gave up then. It was of minimal interest to me in my present post teaching math to students in the age range 15-19. The only thing i discovered was that the student edition of Wikipedia 8 costs less than 150 euros.

I enjoyed much more Dani Novak’s online presentation on integrating math, art, and music. A recording of the event is here. This was a presentation with two-way communication. In also saw the recording of Intellectual consumerism in mathematical learning by  Dr. Maria Droujkova.

This was the task:

Comment on two class member blog posts about their dreams and plans of mathematics education (Week 1, Task 3). In your comments, develop the dreams and plans to address diverse student interests and the approaches they find meaningful. Use particular task example(s) to illustrate the points you make. As always, duplicate comments at your own blog for safekeeping.

Task 3 was: “… In the post, also describe your dreams and plans for mathematics education, in the context of curriculum development.”

The problem is that few, if any (me included), shared their dreams and plans. This illustrates something very important: “Students’ learning, whoever they are, can not be planned in detail beforehand.”

The students may not find the questions or topic relevant to them right now, but it may inspire in them thoughts and questions they want to pursue instead. If the educational setting does not allow for this freedom, boredom and lack of meaning sets in and they fall asleep, that is they disconnect.

‘Having said that’ (with thanks to Larry David) I found Ana and Bernadette’s comments interesting. Ana discusses Conrad Wolfram’s TED talk and says: “Coming up with interesting, real world questions is the biggest challenge for a teacher.” I want to ask: is this the teacher’s task? The tasks should be interesting for the students so shouldn’t they tell the teacher and not the other way around? This is obvious if the students are us taking this online course with Maria, but why should it be different for younger students?

I am not saying it is easy to find out what interests each student have, but I see that as the teacher’s challenge as opposed to guessing what might interest them.

I see the biggest obstacle to meaningful math lessons the prescribed curriculum.

SIMPLICIO:  So  we’re  supposed  to  just  set  off  on  some  free-form  mathematical excursion, and the students will learn whatever they happen to learn?

SALVIATI:  Precisely.    Problems  will  lead  to  other  problems,  technique  will  be developed as  it becomes necessary, and new topics will arise naturally.

And  if  some  issue  never  happens  to  come  up  in  thirteen  years  of schooling, how interesting or important could it be?

SIMPLICIO:  You’ve gone completely mad.

From A Mathematician’s Lament by Paul Lockhart.

Bernadette asks: “why do so many flock to the technology… and not the psychology of our subject?” I have taught for 25 years in 15 different schools in several countries and my sad observation is that math teachers are not given time to reflect on what they are doing. At Google I am told they have 20% of their time to work on what they fancy. Imagine if teachers had one day per week set aside for reflections and experiments!

In short, the math teachers I have worked with have not flocked anywhere, neither to technology nor to psychology. They have just tried to keep their head above the water. Which has been totally acceptable in schools that is doing the same, with no educational philosophy at all.

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.

I just got a calculator, but no accompanying manual.  If you are good at using parentheses you can do the most complicated calculation in one go. If you are not so good, the memories A, B, … come in handy. But how do you store a number in memory A, and how do you recall what you have stored?

A student of mine showed me how:

To store in A: press the buttons SHIFT STO A.

To recall the content from A: press ALPHA A.

Week 1

Explore the InterActivity: Find the Largest Triangle in the Circle

Challenge: Make a triangle in the circle with the largest area.

more soon …