The Hunt Institute has produced some videos featuring the lead writers (including me) talking about various aspects of the standards. Most readers of this blog probably already know everything that is in them, but you might find them useful in educating others about the standards.
I am a secondary-level New Mexico math teacher who has been completing a large, instructional math database for my students who often are ill-prepared for secondary math. My current project involves transitioning between the NM standards and the Common Core Standards. But after just spending a few days on this transition, three problems have arisen. First, there appears to be no mention of “divisiblity” in the Standards. Second, no mention of “improper fractions.” Third, and most important, adding and subtracting fractions with unlike denominators are in grade 5 of the NF cluster but the only reference to least common denominators is in grade 6 of the NS (the Number System) cluster. What gives here? Does anybody know? Is it an oversight from the original committee and will there be a corrected version of the Standards?
The omission of improper fractions and the downplaying of least common denominators are both intentional design features of the Standards, not oversights. The fractions progression explains in more detail, but briefly: the concept of an improper fraction is not an important mathematical concept, and the term can lead children to think that fractions are supposed to be less than one, and that there is something wrong with fractions bigger than one. The Standards aim to develop a unified notion of fractions that does not make this artificial distinction; you’ll notice that fractions larger than one are introduced from the very beginning. Similarly, insisting on finding a least common denominator, rather than simply using the product of the denominators,
is a waste oftakes up precious time in the curriculumwhichand gets in the way of conceptual understanding of fraction addition. With the notion of fractions as multiples of unit fractions, it is natural in adding them to find a common unit by subdividing each unit fraction into the number of parts indicated by the denominator of the other fraction. This is an important but not an easy insight; it only muddies the waters to insist that the common unit be the most efficient one. Worse, students can end up believing that least common denominators are somehow an essential component of fraction addition, with the result that some of them end up not being able to add fractions at all.The matter of the word “divisible” is somewhat different. Although that particular word is not used, the concepts involved in its definition are important in the Standards. Students learn to understand a division as the answer to an unknown multiplicand problem (that is 10÷5 as the answer to 5x? = 10). They initially work with whole number division with whole number answers, and then move on to fractional answers once they have an understanding of the fraction a/b as a÷b. They also learn about the concept of quotient and remainder. These ideas are developed in Grades 3–5. In describing the development we didn’t find a need for the word “divisible”, but I don’t see that as a prohibition against defining the word in the classroom if the occasion arises; the ideas needed to understand it are certainly there.
[Updated 12/16/11, 11:43 am MST]
First of all, I greatly appreciate your very, very quick response to my statement of the quandary I found myself in. And I read and reread it so as not to miss anything.
A thought, though, from a secondary teacher who has taught basic fractions to 2/3 of my at-risk students because they, for the most part, have had absolutely NO understanding of the subject: they actually enjoy finding LCM’s even when the numbers are somewhat dire (12 and 16, for example) and they somewhat easily progress to fully using this concept. That it is a “waste of time” is something difficult for me to comprehend when I see (and hear of, second-hand) that they “finally” understand fractions!! You have to understand that the absolute majority of students I come across have only a 3rd to 4th grade ‘education’ and introducing these concepts (which is a curriculum necessarily designed by myself alone as no other seemed to adequately present itself ) has given (most of) them a new sense of what it means to be mathematically literate.
Lastly, I hope there is room in the Common Core state of mind to have reasonable people disagree with one another.
Again, I greatly appreciate your response and hope that we can have an ongoing dialogue as I meet up with more issues.
I agree that an enjoyable mathematical activity is not a waste if time, particularly with at-risk students. It would have been more accurate to say that it takes up time in the curriculum (and I have edited my comment to say that). There is not enough time for all the things we might want to do; the point of the Common Core is to agree on a focused set of things and try to do them well. One of the goals of the fraction progression is to prepare students for algebra. If students get in the habit of adding fractions using the product of the two denominators (rather than the least common denominator), then they might be better prepared to manipulate algebraic expressions like
.
I’m always happy to encounter and discuss disagreements. But your argument here is not really with me, but with the document itself. The Common Core does not in fact support a heavy emphasis on least common denominators. This doesn’t mean you have to stop teaching the way you have been teaching, but it does mean that you cannot both do that and at the same time claim to be following the Common Core. The benefit of the Common Core lies in the shared knowledge and tools that become available when everybody is on the same page; we can’t achieve that benefit if we all continue to do our own thing.
Thanks again for the reply as I am working on this project as we correspond.
My intention is to adhere to the outline of the CORE concepts you discuss (with over 1000 topics already created with 30 or so problems apiece) with the added intention of incorporating new topic areas that are spot-on regarding the Common Core approach but I can’t seem to agree with the statement “it does mean that you cannot both do that and at the same time claim to be following the Common Core.” when, in my view, I am adding/enhancing to it and not actually disavowing it.