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.