Singapore Mental Math

I am exploring the Bar Model at the moment and have bought some books to help me get to grips with it.  Most of these books are very interesting and contain illuminating ideas.  When I have digested them I will post a list of the ones I found most helpful.

While browsing through books on Amazon I also bought a Singapore Math ‘Mental Math’ book. This is very different from the bar model materials.

The book describes itself as containing ‘challenging activities based on the world-renowned Singapore Math curriculum’.  It says it includes ‘must-know strategies for solving problems quickly and accurately’.  It states: ‘To help students build and strengthen their mental calculation skills, this book provides strategies that will benefit students as they learn tips to solve math problems quickly and effectively.  After acquiring such invaluable skills, students can apply them to their future, real-life experiences with math, such as in shopping and banking.’

There are 52 strategies in the book (one for each week of the year) and many of them are quite extraordinary. In the rest of this post I will explain one of them and will then discuss what I see as the pros and cons of this approach.  Future blog posts will include other strategies from the book.

Week 24 has this strategy:

The good things about this:

• Pupils will practise following instructions and using an algorithm.
• It might make a good ‘settler’ task at the start of a lesson.
• Some mental maths muscles will be flexed (with lots of adding and doubling going on).

The bad things about this:

• This strategy has very limited application.
• Pupils are being encouraged to learn lots of strategies of questionable usefulness.
• There could be confusion and it could be applied in the wrong setting.
• There is a much easier way to do this!
• There is nothing to explain why this works or to lead pupils to explore it.

Perhaps I am being unfair, and maybe pupils will want to know how and why it works.  Maybe teachers using this resource _do_ explore why this works.  If so then I think this could be a wonderfully positive resource.  For example:

Let’s use some algebra.  Call the two numbers n+1 and n.

We have (n+1)² – n².  We could expand the brackets, to give n² + 2n + 1 – n²

Alternatively we could use the difference of two squares:

This gives us (n+1)² – n² = (n+1 + n)(n+1 - n), which simplifies to (2n+1).

Going back to the original:

The original has (n+ ½ ) multiplied by 2, which is 2n+1.  

The easier way:

But 2n+1 is also (n+1) + n, so this is actually the two numbers added together!  11+10 = 21

An alternative method could involve drawing some diagrams:

This shows that the non-yellow part contains (n+1) + n squares, and properly demonstrates that we are adding the two numbers together.

I prefer the difference of two squares method because it seems to me to be the most easily generalizable:

If the numbers are two apart then we have (n+2)² – n², which is (2n+2)x2

If the numbers are three apart then we have (n+3)² – n², which is (2n+3)x3.

Every time (and using (n+a)² – n²) makes this obvious), we need to add the two numbers and then multiply by their difference. 

I can see some really fertile material here for KS3 and KS4 pupils to explore, so as a starting point this turns out to be an exciting resource.  It also leads back to the important idea that there is no such thing as a ‘good resource’ or a ‘bad resource’, but that it depends on how you use them.