Common Core Math Standards Ignore the Role of Pattern

    

I still cannot decide if I think the new Common Core Math Standards (CCMSs) are the best thing since sliced bread, just abother phase in the development of American Education or an imminent disaster. I've just spent three days at the ATMNE (Association of Teachers of Mathematics of New England) conference listening to a variety of speakers and presenters sharing their ideas and strategies for implementing the standards in classrooms througout New England. What concerns me most is that all the people I respect in the field of maths education all seem to have different views about the CCMSs. Most are passionately against or passionately for them; there seem to be few indifferent opinions. 

The focus on the math practice standards is probably a good change although the recognition of the importance of finding patterns in mathematics seems to be completely missing. Yes, finding structure is included as a practice standard but this is not the same.

For example in one of the presentations on misconceptions in fractions the presenter used an example of comparing the fractions 5/6 and 6/7 I think it was. Children, apparently, gave all soprts of answers and predominantly said they were the same because there was only 1 difference between the numerator and denominator in each fraction. Most teachers suggested converting to decimals or finding the common denominators.

But, if we taught children all about the inherent patterns in math they would be able to compare them easily. 

Look at this pattern; 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, 7/8, 8/9, 9/10. Numerators and denominators increase by 1 and each fraction is getting progressively closer to 1. In other words the bit left out of 1 not included in the fraction is getting smaller; 1/10 is smaller than 1/2, We can use models to show this.

Look at this pattern; 1/2, 2/4, 3/6, 4/8, 5/10, 6/12. All are other names for a half. In each one the numerator is half the denominator; the numeraotr increase by 1 and the denominators increase by 2.

Now look at this pattern; 1/3, 2/6, 3/9, 4/12, 5/15, 6/18. All are other names for 1/3 and in each the numerator is a third of the denominator with numerators increasing y 1 and denominators increasing by 3.

Just for fun, look at the half pattern again and find a fraction between 1/2 and 2/4. How about one and a half thirds? 11/2 is between 1 and 2, and 3 is between 2 and 4. It's actually 3/6.  Now try the fraction between 2/4 and 3/6.   

Fractions, aaaaahh fractions

A mathematician saw an advertisement in a newspaper for a vacuum cleaner that was so good, it said, that it cuts your work in half. That sounds like a great deal, she thought, I must buy two of those. Tada!

Would you rather have half the money I have in my right hand or a quarter of the money I have in my left hand?

We use the language of fractions constantly in our lives from the time we are old enough to ask for half a cup of orange juice or asked to share a candy bar with a brother or sister so that we have half each. Ask any six year old how old she is and you will most likely get the reply "six and  half". Everything is just fine until we come across fractions in school and then all hell breaks loose and we leave common sense at the classroom door.

Remember the rigmarole we were taught to find 1/2 divided by 1/4? I seem to remember "Change the sign and flip the second fraction" as being the most common piece of nonsense we were taught in 4th grade. We never questioned the wisdom of this piece of sage advice because we never had a clue why it worked. It was part of the magic of math; it just worked when we had to use it on an exam question. Never mind the fact that we had no idea when to use it; "was it half of a quarter or how many times does a half go into a quarter or ........?"

How much easier life is now that we use mathematical tools such as the fraction strips in the picture. 1/2 divided by 1/4 actually means how many 1/4s in a 1/2 as in "how many 1/4s in the first 1/2 of a football game?". How many yellow pieces fit in one pink piece above?

As for the 'which handful of money would you rather have' question; it depends on how much money I have in each hand. The size of any fraction depends on the size of the whole to which it refers.

This is the topic of my math class this afternoon.