[Maths Education] Mathematical Journalling

     Nowadays, I see some schools / textbooks asking students to search the internet and to write on a certain problem on their “mathematics journal”.  It's so very guided.  It's so artificial.  The questions should come from the learners themselves, out of their own curiosity.  The learn then seeks to answer their own questions.  The journal can serve as to document and summarise their process of learning.

     The best maths journals are self-initiated.  Great mathematician Karl Friedrich Gauss and renowned scientist Richard Feynman kept math journals on their own accord, not because some teacher told them to do it. 

     When I was a student, I borrowed books from the National Library on things out of the normal curriculum.  I kept notes of things I learned.  I also did my own investigations.  I accidently discovered quadratic equations when I was in Primary 4.  I read guidebooks, asked my friend's elder brothers and sisters, my Chinese teacher (!) and other people to find out more.  I did not like factorisation by trial-and-error.  Neither did I like completing the square nor using the quadratic formula.  So I did my own research to find a sure-fire way to factorise without trial-and-error.  I finally managed to find a way, but my method had an uncanny similarity to the quadratic formula.  It was a Pyrrhic victory, but it was fun!  I thoroughly enjoyed it.

     If students need to be told or goaded to write mathematics journals, then we as educators need to ask ourselves:  Why?  What is their conception of mathematics and education?  What experiences have they gone through that lead them to these beliefs?

     Some food for thought, eh?

[Maths_Education] The Need to Harness and Transcend Technology

Problem / Question

Help me please!  My teacher needs it tomorrow! How many millilitres are there in 3.4 litres?

Answer

     LOL!  Well, if she really needs it, ask her to type the question into Google!

Remarks

     Actually, this is not a joke.  Many easy questions in school mathematics are now answerable by Google.  In fact, Wolfram Alpha is able to answer mathematically more difficult but routine questions.  Wikipedia and YouTube are also useful for learning mathematics.  There are many other good resources available on the Internet and public libraries.  The sad thing is,

1)	It seems that our current generation of kids views homework as a chore to be done for the teacher, not as an experience to be used for their own learning.  OK, maybe the homework task should have been designed better, to ask non-Googleable questions, but educators need to be aware of that is happening to our kids.
2)	Our kids do not know how to choose and use the abundantly available technology and resources
3)	Neither are they taught how to do this in school (do the teachers know it themselves?)
4)	This type of question merely targets the lower levels of Blooms Taxonomy
5)	Even for these easy questions, children are unable or unwilling to make the effort to find their own answers, and how are they to engage in Higher Order Thinking, creative thinking, reasoning etc.?

     As technology evolves and improves and replaces many jobs, and as our children de-evolve and slacken, when this generation grows up they will not only face an even more challenging environment for their careers than today, they may not have the right values, attitudes and dispositions for living life.
     The purpose of learning mathematics in school should be to learn how to think and to serve others.  Human students must be educated to use technology and go beyond technology, to seek answers and to help other people instead of merely relying from other people for “help”.  This is one of the reasons why an identity (“learning to be a type of person”) approach to learning mathematics is important.

[Maths_Education] Is Learning Mathematics a "Performance"?

Article
The Math-Class Paradox


My Comments
     This is an excellent article addressing the issue of mathematics education in schools today.  School "mathematics" barely touches even the tip of my iceberg.  Is the learning of mathematics merely a matter of performing under time pressure on tests and exams (and that puts many students off)?  What about exploration, creativity, asking questions and discovering connections ... etc?
     The thought provoking article is written by Jo Boaler, one of my favorite professors of mathematics education.  It is definitely worth your while reading it!

Mathematics Education expert speaks

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Research tell us what to do with mathematics education crisis

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[OlymPri20150402RLD] Peter, James and the Dog

 Question

Introduction

     This is a cute primary school olympiad type of question.  I vaguely remember reading a similar quiz question from Readers’ Digest (?) many years ago about some bumble bee flying between two trains going towards each other.  Or something like that.

     If you try to solve this using “advanced mathematics” like using the sum of two infinite geometric series you can get the answer, but it is quite a swirling mess, like this:-

Another way to slice the dice

     Is there a short cut?  Yes!  That requires thinking about the problem in another way.  Notice that the dog’s speed is the sum of Peter’s and James’ speeds.  Imagine ... if James is not moving, but Peter is running at  3 ms^-1,  from Peter’s point of view the Earth would be pushed backwards at 3 ms^-1  and James would appear to be going towards him at  3 ms^-1.  But James is running towards Peter at  2 ms^-1, so from Peter’s point of view, it seems that James is coming towards him at  5 ms^-1.  Likewise, from James’ point of view, Peter appears to be coming at him at  5 ms^-1.  This is the concept of relative speed.  Notice also that the relative distance (gap) between James and Peter is reducing at this speed.  This is because at Peters’ end the gap is reduced at 3 ms^-1  and at James’ end the distance is reduced by  2 ms^-1, giving a total gap-reduction speed of  5 ms^-1.  With these perceptive observations, the answer falls straight out.

Solution

     Since the dog’s speed (5 ms^-1) is the sum of Peter’s speed (3 ms^-1) and James’ speed (2 ms^-1), it is always covering a distance at a speed which is the same as the speed of the closing of the gap between Peter and James.  Hence the total distance travelled by the dog must be the same as the initial gap, which is  1 km. 

Moral of the Story

     Sometimes, you do not need advanced maths, but acute observations.

When two entities are moving towards each other, their relative speed is the sum of their speeds.  This is also the same as the rate at which the relative distance (gap between the two) is closing.

Suitable Levels
* Primary School Olympiad
* anyone who is interested in creative maths problem solving