2+2 != Maths

October 30, 2011

Division as sharing helps with fractions.

Filed under: teaching — Numbat @ 15:50
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Hi All,

Thanks to a number of twitter references of late I have managed to really solidify my thinking on the concept of division.  I had never really given it much thought as division came a little too easily to my mind, however there are two very subtly different concepts of division.

The first, which is commonly referred to as “chunking”, involves breaking an amount up into groups of a certain size.  So, for example, 10 ÷ 5 becomes “break 10 into groups of 5”.  This is what I have found to be the more prevalent way of thinking about division amongst high school students.

The second method doesn’t appear to have a common name, and so I am going to call it the “sharing” method.  This involves sharing the amount between a given number of groups.  So, the same sum, 10 ÷ 5 becomes “share 10 items amongst 5 groups”.

The difference is subtle, and best represented graphically, as follows.

1) 10 ÷ 5 : Thought as 10 divided into groups of 5. How many groups of 5?

2) 10 ÷ 5 : Thought as 10 shared between 5 groups. How many does each group receive?

When teaching division as the opposite of multiplication, it becomes easy for students to concentrate on the “chunking” idea of division.  10 ÷ 5 = 2 because 2 x 5 = 10, or 2 groups of 5 equals 10. As stated, I have found this to be the more prevalent way of thinking amongst high school students.  However, I think the “sharing” method is probably the more “natural” way to think of division.

If I were to give 6 sweets to a group of three children, they would very easily calculate that they would each receive two sweets.  6 ÷ 3 = 2.  However, if I handed the same group of three an unknown quantity of sweets, they would still be capable of dividing them equally.

I very much doubt that would add up the sweets then divide by three. No, they would simply dole them out one at a time to each and end up with an equal share.

The two concepts of division become very important when it comes time to introduce fractions to students.  When presented with the fraction ¼, students who’s primary concept of division is chunking find the image of ¼ difficult to understand.  How can I split 1 into groups of 4?  It just doesn’t compute.

However, students who see division as “sharing” find it easier to compute that fraction.  “Share 1 pie among 4 students” is easier – simply cut the pie into 4 equal parts and dole them out.  This is what they’ve been doing as kids all along.



September 9, 2011

My first problem

Filed under: problem solving — Numbat @ 11:01
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Hi All,

I awoke a little “mathy” today.The first order of business most days for me is to check the blogs and twitter feeds that I follow.  It kickstarts the brain quite nicely.

I had to watch the following video twice.

The first time I watched, just as he got into the 2nd example (26 seconds into the video), my brain suddenly kicked into gear and screamed “No way!”.  As my early morning brain grappled with what I was seeing and James continued on with his explanation, it took a few minutes for me to realise the “proof” was staring me right in the face.

I immediately wondered if my students would see through these examples as quickly (or slowly as the case may be) as I did.  So there is the basis of my problem.  The first problem I have attempted to “create” I might add.  Please feel free to let me know what you think and make suggestions for improvement in the comments.

The problem:

Watch the first 45 seconds of the above video, then ask your students to either prove or disprove the method. You could make it a little harder (or perhaps easier) if you ask them to only use the 4 examples shown.


Thanks go to @jamestanton for the video and @ddmeyer for the inspiration.

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