I just finished posting an opinion to a thread on FB that had this graphic with it. It began here:
Do you know how to do Common Core math? Confusion over the standards has some calling for their removal. NBC’s Rehema Ellis reports tonight.”
One reader had addressed me specifically in that thread, asking for my opinion, but as I responded to her under my personal FB account and not my Focus on Math account, I decided to copy and paste my thoughts here on the blog (and thus to my FoM FB account) so they are officially “on the record”. Here is my post:
Louise Cook, This is about learning to think about numbers deeply. Most adults, if asked to add or subtract two two-digit numbers in their heads, cannot do it. They have to to the “carrying” or “borrowing” formula (on paper usually, or with great difficulty in their heads), and have no alternative ways to think about the numbers. If children are truly exploring numbers regularly it is AMAZING how they learn to manipulate numbers. More importantly, they understand deeply about the number system and why things work the way they do. The true mathematics is not just in memorizing methods that spit out an answer, but understanding the concepts that under lie those methods. Adults may be able to divide fractions (say, divide 1/2 by 3/8 using the “invert and multiply” method, but very, very few understand and can explain WHY they get an answer of 1 1/3. They think it just “magically” appears as an answer, but they don’t know why. I think that moving numbers around in operations (adding, subtracting, multiplying, dividing) without understanding why the methods work is like allowing a student to “read” without having any comprehension of what the words said. In fact, is it really reading without comprehension? Part of the problem is that we have been teaching math in the “spit out an answer way” for so long, that we believe it is right just because it feels comfortable. (I will also add that many teachers have been asked to teach for deeper understanding, such as with multiple strategies, without having developed an understanding for themselves. That is rather like trying to teach a foreign language that you don’t speak yourself — a difficult task to say the least. No wonder is does not work!) It was George Polya, a pioneer in problem solving, who said, “better to solve one problem five ways that five problems one way.” (end of that post)
I believe deeply in letting children really think about math, not spitting out a page of fifty numerical problems on a page without a context, ones they are expected to solve in a prescribed manner. If this method was doing such a great job we would have all of Canada and the US loving math and feeling confident about it. Instead, the vast majority of folks will say they hated math, were not (and are not) good at it, etc. Math usually wins as the “least-liked subject in school”. Oh, dear, I am still fired up about this and could go on and on, but it is late and I must stop. I am happy to continue chatting about this.
Mathematically yours,
Carollee I just finished posting an opinion to a thread on FB that had this graphic with it. It began here:
Do you know how to do Common Core math? Confusion over the standards has some calling for their removal. NBC’s Rehema Ellis reports tonight.”
One reader had addressed me specifically in that thread, asking for my opinion, but as I responded to her under my personal FB account and not my Focus on Math account, I decided to copy and paste my thoughts here on the blog (and thus to my FoM FB account) so they are officially “on the record”. Here is my post:
Louise Cook, This is about learning to think about numbers deeply. Most adults, if asked to add or subtract two two-digit numbers in their heads, cannot do it. They have to to the “carrying” or “borrowing” formula (on paper usually, or with great difficulty in their heads), and have no alternative ways to think about the numbers. If children are truly exploring numbers regularly it is AMAZING how they learn to manipulate numbers. More importantly, they understand deeply about the number system and why things work the way they do. The true mathematics is not just in memorizing methods that spit out an answer, but understanding the concepts that under lie those methods. Adults may be able to divide fractions (say, divide 1/2 by 3/8 using the “invert and multiply” method, but very, very few understand and can explain WHY they get an answer of 1 1/3. They think it just “magically” appears as an answer, but they don’t know why. I think that moving numbers around in operations (adding, subtracting, multiplying, dividing) without understanding why the methods work is like allowing a student to “read” without having any comprehension of what the words said. In fact, is it really reading without comprehension? Part of the problem is that we have been teaching math in the “spit out an answer way” for so long, that we believe it is right just because it feels comfortable. (I will also add that many teachers have been asked to teach for deeper understanding, such as with multiple strategies, without having developed an understanding for themselves. That is rather like trying to teach a foreign language that you don’t speak yourself — a difficult task to say the least. No wonder is does not work!) It was George Polya, a pioneer in problem solving, who said, “better to solve one problem five ways that five problems one way.” (end of that post)
I believe deeply in letting children really think about math, not spitting out a page of fifty numerical problems on a page without a context, ones they are expected to solve in a prescribed manner. If this method was doing such a great job we would have all of Canada and the US loving math and feeling confident about it. Instead, the vast majority of folks will say they hated math, were not (and are not) good at it, etc. Math usually wins as the “least-liked subject in school”. Oh, dear, I am still fired up about this and could go on and on, but it is late and I must stop. I am happy to continue chatting about this.
Mathematically yours,
Carollee 
You have done an excellent job explaining why the conceptual understanding is so vital. I especially like your analogy about teaching a language you don’t speak.
I really loved your statement:
“If this method was doing such a great job we would have all of Canada and the US loving math and feeling confident about it.”
My feelings exactly.
I’ve been following the math wars in the United States, and I am ambivalent about the CCSS, not because of the standards, but because it seems to be ushering in a regime of extremely high-stakes testing. However, the reaction against the math content of the CCSS reminds me of what has happened in Alberta with the unfortunate back-to-basics push. We have some superb teachers in Alberta who have managed to teach not only the basics but a great deal more.
Ted Lewis
http://www.mathinautumn.blogspot.ca
I don’t know where the graphic came from but in itself it shows a complete misunderstanding of the intentions of the Common Core. Ah, well.
I love your statement about fraction division. I work with a group of well educated teachers and 5 years ago only 10% of them could write a story problem about fraction division. The Common Core is so misunderstood!